Empirico-Statistical Analysis of Narrative Material and
its Applications to Historical Dating
Volume I
Empirico-Statistical Analysis of
Narrative Material and
its Applications
to Historical Dating
Volume I:
The Development of the Statistical Tools
by
A.T. FOMENKO
Department of Geometry and Topology ,
Faculty of Mathematics,
Moscow University,
Moscow, Russia
KLUWER ACADEMIC PUBLISHERS
DORDRECHT / BOSTON / LONDON
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 0-7923-2604-0 (Volume I)
ISBN 0-7923-2606-7 (Set of 2 Volumes)
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Translated by O. Efimov
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Contents
Foreword xi
Preface xiii
Chapter 1. Problems of Ancient and Medieval Chronology 1
§1. The Global Chronological Diagram of Ancient and Medieval
History 1
1.1. The moon’s elongation and R. Newton’s conjecture 1
1.2. The Dark Ages and the Renaissance epochs 2
1.3. How to substantiate ancient chronology 3
1.4. Statistical dating methods: new possibilities 5
1.5. The duplication effect in ancient history and chronology 7
1.6. The global chronological diagram and the “modern
textbook” of ancient and medieval history 8
1.7. The “modern textbook”, a composition of four
identical pieces 10
1.8. Certain corollaries and interpretations 11
1.9. What is to be done with the moon’s elongation? 12
References 13
§2. Computation of the Second Derivative of the Moon’s Elongation
and Statistical Regularities in the Distribution of the Records of
Ancient Eclipses 15
2.1. Parameter D " and R. Newton’s paper “Astronomical
evidence concerning non-gravitational forces in the
Earth-Moon system” 15
2.2. Available observations of ancient solar and lunar eclipses 17
2.3. A method of formal astronomical dating 18
2.4. The effect of shifting the dates of eclipses forwards 19
2.5. An example: three eclipses of Thucydides 20
2.6. An example: the eclipse described by Livy 22
2.7. An example: the eclipse described by Livy and Plutarch 23
2.8. An example: the evangelical eclipse described in the New
Testament in connection with the Crucifixion 24
2.9. The oscillation of a new graph of D ft about one and the
same value. No nongravitational theories are necessary 25
v
VI
Contents
2.10. Three rigid “astronomical shifts” of ancient eclipses 28
2.11. The complete picture of astronomical shifts 31
2.12. The coincidence of the astronomical shifts with the three
basic chronological shifts in the global chronological
diagram 31
§3. Traditional Chronology of the Flares of Stars and the Dating
of Ancient Horoscopes 32
3.1. Ancient and medieval flares of stars. The star of Bethlehem 32
3.2. Astronomical dating of ancient Egyptian horoscopes 34
3.3. Astronomical dating of the horoscope described in the
Book of Revelation 35
References 37
Chapter 2. New Statistical Methods for Dating 39
§4. Certain Statistical Regularities of Information Density
Distribution in Texts with A Scale 39
4.1. Text with a scale. The general notion 39
4.2. Information characteristics (i.e., informative functions) of
a historical text. Volume function, name function, and
reference function 40
4.3. A theoretical model describing the distribution of
local maxima for the volume function of a historical text.
Primary stock. The information density conservation law 42
4.4. The correlation of local maxima for the volume graphs of
dependent historical chronicles. The surviving-stock graph 43
4.5. Mathematical formalization. The numerical coefficient
d(X, Y), which measures the “distance” between two
historical texts X and Y 44
4.6. Mathematical formulas for computing d(X, Y). Mathematical
corrections of the maxima correlation principle 47
4.7. Verification of the maxima correlation principle against
concrete historical material 49
4.8. A new method for dating historical events. The method
of restoring the graph of the primary and surviving
information stock 52
4.9. The discovery of dependent (parallel) historical epochs
traditionally regarded as different 53
4.10. The dynasty of rulers and the durations of their reigns as an
important informative function 54
4.11. Frequency distribution of the rules of kings who lived
from a.d. 1400 to 1800 and from 3000 B.c. to a.d. 1800 55
4.12. The concept of statistically parallel historical texts and epochs 57
4.13. The “written biography” or enquete-code of a historical
character 57
Contents vii
4.14. A method of comparing the sets of informative functions
for two historical epochs 60
4.15. A computational experiment 61
4.16. The remarkable decomposition of the global chronological
diagram into the sum of four practically indistinguishable
chronicles 62
References 66
§5. A Method of Duplicate Recognition and Some Applications to
the Chronology of Ancient Dynasties 67
5.1. The process of measuring random variables 67
5.2. The distance between two random vectors 67
5.3. Dynasties of rulers. The real dynasty and the
numerical dynasty. Dependent and independent dynasties.
The small- distortion principle 68
5.4. Basic errors leading to controversy among chroniclers as to
the duration of kings’ rules 69
5.5. The experimental frequency histogram for the duration of
the rules of kings 70
5.6. Virtual dynasties and a mathematical model for errors made
by the chronicler in measuring the rule duration 70
5.7. The small-distortion principle and a computer experiment 72
5.8. Pairs of dependent historical dynasties previously regarded
as independent 73
5.9. The distribution of dependent dynasties in the “modern
textbook” of ancient history 73
5.10. Dependent dynasties in the Bible and parallel with European
history 74
References 75
§6. A New Empirico-Statistical Procedure for Text Ordering and Its
Applications to the Problems of Dating 76
6.1. The chapter generation 76
6.2. The frequency-damping principle 76
6.3. The method of finding the chronologically correct order of
chapters in a historical chronicle 77
6.4. The frequency-duplicating principle and the method of
duplicate recognition 79
6.5. The distribution of old and new duplicates in the Old and
New Testament. A striking example: the Book of Revelation 81
6.6. Duplicates of epochs in the “modern textbook” of ancient
history 83
References 87
Contents
viii
Chapter 3. New Experimental and Statistical Methods for
Dating Events of Ancient History, and Their Applications
to the Global Chronology of Ancient and Medieval History 88
§7. Introduction. N.A. Morozov and Modern Results 88
§8. Problems of Historical Chronology 89
8.1. Roman chronology as the “spinal column” of European
chronology 89
8.2. Scaliger, Petavius, Christian chronographers and secular
chronography 90
8.3. Questioning the authenticity of Roman tradition.
Hypercriticism and T. Mommsen 93
8.4. Difficulties in the establishment of Egyptian chronology 94
8.5. Competing chronological versions. De Arcilla, J. Hardouin,
I. Newton and R. Baldauf 96
8.6. Tacitus and Bracciolioni. Cicero and Barzizza 97
8.7. Vitruvius and L. Alberti 98
8.8. “The chaos of medieval datings” (E. Bickerman). Medieval
anachronisms and medieval concepts of time 99
8.9. The chronology of the biblical manuscripts. L. Tischendorf 102
8.10. Vowels in ancient manuscripts 103
8.11. Traditional biblical geography 104
8.12. Problems of geographical localization of ancient events 105
8.13. Modern analysis of biblical geography 107
8.14. Ancient originals and medieval duplicates. Anachronisms
as a common feature in medieval chronicles 110
8.15. Names and nicknames. Handwritten books 112
§9. Astronomical and Mathematical Analysis of the Almagest 113
9.1. Morozov’s analysis of the first medieval editions of the
Almagest 113
9.2. On the statistical characteristics of the Almagest. The
structure of the star catalogue 115
9.3. The accuracy of the Almagest’s star coordinates 119
9.4. The problem of dating the Almagest from the individual
stars’ proper motion 120
9.5. Halley’s discovery of the stars’ proper motion and the
Almagest 128
§10. Archaeological Dating Methods 131
10.1. Classical excavation methods 131
10.2. Numismatics 132
10.3. The dendrochronological method 133
10.4. The radiocarbon method 133
§11. Astronomical Dating. Ancient Eclipses and Horoscopes 136
§12. New Experimental and Statistical Methods of Dating Ancient
Events 140
Contents
IX
12.1. Introduction
12.2. Volume graphs for historical chronicles. The maximum
correlation principle. Computational experiments and
typical examples
12.3. Method of recognition and dating the dynasties of ancient
rulers. The small-distortion principle
12.4. The frequency-damping principle. A method of ordering
texts in time
12.5. Applications to Roman and Greek history
12.6. The frequency-duplication principle. The duplicate-
discovery method
12.7. Statistical analysis of the complete list of all the names
mentioned in the Bible
12.8. Statistical analysis of the complete list of all parallel
passages in the Bible
12.9. Duplicates in the Bible
12.10. The enquete-code or formalized “biography” method
12.11. A method for the chronological ordering of ancient maps
§13. Construction of the Global Chronological Diagram and Certain
Results of Applying the Dating Methods to Ancient History
13.1. The “textbook” of ancient and medieval history
13.2. Duplicates
13.3. Dependent dynasties
13.4. The agreement of different methods
13.5. Three basic chronological shifts
13.6. Biblical history and European history
13.7. The beginning of “authentic” history in circa the 10th
century A.D.
13.8. The chronological version of Morozov and the author’s
conception
13.9. The confusion between the two Romes
13.10. A universal mechanism which could lead to the chroniclers’
chronological errors
13.11. Scaliger, Petavius, and the Council of Trent. Creation of
traditional chronology
§14. The “Dark Ages” in Medieval History
14.1. Medieval Italy and Rome
14.2. Medieval Greece and Athens
14.3. The history of religions
14.4. Indian history and chronology
References
Index
203
Foreword
Today the methods of applied statistics have penetrated very different fields of
knowledge, including the investigation of texts of various origins. These “texts”
may be considered as signal sequences of different kinds, long genetic codes,
graphic representations (which may be coded and represented by a “text”),
as well as actual narrative texts (for example, historical chronicles, originals,
documents, etc.) . One of the most important problems arising here is to recognize
dependent text, i.e., texts which have a measure of “resemblance” , arising from
some kind of “common origin” . For instance, in pattern-recognition problems,
it is essential to identify from a large set of “patterns” a pattern that is “closest”
to a given one; in studying long signal sequences, it is important to recognize
“homogeneous subsequences” and the places of their junction. This includes, in
particular, the well-known change-point problem, which is given considerable
attention in mathematical statistics and the theory of stochastic processes.
As applied to the study of narrative texts, the problem of recognizing depen-
dent and independent texts (e.g., chronicles) leads to the problem of finding texts
having a common source, i.e., the same original (such texts are naturally called
dependent ), or, on the contrary, having different sources (such texts are natu-
rally called independent ). Clearly, such problems are exceedingly complicated,
and therefore the appearance of new empirico-statistical recognition methods
which, along with the classical approaches, may prove useful in concrete studies
(e.g., source determination) is welcome.
The present book by A.T. Fomenko, professor of pure mathematics, is mainly
aimed at developing such new methods to be applied to recognizing dependent
and independent narrative texts and dating them (with respect to texts with
known reliable dates).
The author proposes a new approach to the problem of recognizing dependent
and independent narrative (historical) texts based on the several new empirico-
statistical models (regularities) which he has discovered during his extensive sta-
tistical emperiments involving various quantitative characteristics of concrete
texts, chronicles, originals, and so forth. Verification of these models (statisti-
cal hypotheses) on concrete chronicles confirmed the efficacy of the models and
made it possible to put forward new methos for dating texts (more precisely, for
dating events described in the texts).
The approach proposed in this book is nonstandard and requires attention
and diligence on the part of the reader to new and probably unfamiliar logical
constructions. At the same time, the basic ideas of the author seem quite natural
xi
xii Foreword
from the viewpoint of modern mathematical statistics and can easily be included
in the conceptual system of applied statisticians.
The author’s scientific results and ideas are very interesting, and perhaps al-
ready today we may speak of the appearance of a new (and rather unexpected)
scientific trend in applied statistics whose development is of undisputed impor-
tance. This book is a result of a tremendous amount of work done by the author
and his colleagues, most of whom specialize in mathematical statistics and its
applications.
Since the book is devoted to problems at the interface of several branches of
science, the necessity of establishing contact between people of different profes-
sional backgrounds becomes obvious. Many concepts and terms customary to
specialists in one branch require translation into the language of specialists in
another field of research. This should be borne in mind by representatives of both
the natural sciences and the humanities. Such “difficulties of communication”
are typical and must be successfully overcome with any mixed group of scientists
working on joint problems. One may hope that many of the readers of this book
will join to form such an interdisciplinary group in order to successfully continue
the studies started here by this well-known mathematician.
Along with developing new empirico-statistical methods for dating events, the
book also includes applications to the problem of modern scientific argumen-
tation of the chronology of past events. One should clearly distinguish between
the main statistical result obtained by the author (namely, the layered structure
of the global “chronological map” and its representation as a “sum” of four lay-
ers) and its various interpretations and substantiations. Stating hypotheses and
providing substantiations of results are beyond the scope of exact mathematical
knowledge, and therefore special care should be taken in formulating conclusions
concerning the possible structure of a new “statistical chronology of antiquity” .
The author has repeatedly insisted upon the necessity of critical analysis and
upon distinguishing between strictly established facts and hypotheses or inter-
pretations concerning these facts.
The concepts proposed by the author are new, sometimes unexpected, and
deserve extensive and thorough investigation.
The book is written at a high scientific level, it is a unique phenomenon in the
scientific literature in the field of applications of mathematical statistics, and
the reader will not remain indifferent to it. The book also enables us to get to
know the engaging personality of its author — mathematician and investigator
of history ....
I hope that, after perusing the first few pages of the book, the reader will be
intrigued to read to the end with unabating interest. He will, at the very least, get
to know an interesting set of scientific problems and, perhaps, will even engage
in further investigations in this new and promising field of science.
Albert N. Shiryaev
President of the International Bernoulli Society for
Mathematical Statistics and Probability Theory
Preface
This book presents new empirico-statistical methods for the discovery of de-
pendences between texts, on which we base our dating methods. As one of
various possible applications, the datings (or dates) of certain events described
in ancient and medieval texts are analyzed.
The problem of recognizing dependences (and dependent texts) arises in
many branches of applied statistics, linguistics, physics, genetics, and so forth.
For example, as applied to source research, the discovery of dependent texts
with a common primary source or original (which may not have survived) is
of considerable interest. On the other hand, it is useful to have an idea which
texts may be called independent or are based on substantially different pri-
mary sources and archival data. Meanwhile, the concept of text itself can be
treated extremely differently. We can consider a sequence of symbols, signals,
codes (of various kinds; for example, genetic codes in DNA chains) as a text,
where the general problem in the search for “dependent texts” consists of find-
ing “similar” portions in a given long signal sequence, i.e., textual fragments
“duplicating” each other.
Today, there are many methods for finding dependences of this sort. We sug-
gest certain new empirico-statistical procedures which can prove useful both
in analyzing narrative texts (such as annals and chronicles) and in studying
biological codes to find so-called homologous fragments, and so on.
For the reader’s convenience, we divide the contents into several “topics”,
which may be helpful in getting oriented in the material and in separating
reliable statistical evidence from hypotheses. This devision is arbitrary in the
sense that the highlights listed in the following are intimately related to each
other. Therefore, it would be more correct to speak of the book’s “fibres”,
rather than of its parts. The book’s chapters receive different emphasis, and
I will briefly describe this accentuation here. I hope that the reader will be
able to relate each fragment of the book to some particular “fibre”, and, in
particular, to make out the author’s attitude toward each fibre.
The first fibre. The problem of discovering statistically covert dependences
and dependent texts is solved, to which purpose a number of new statistical
models (or hypotheses) are formulated. They are then checked against suffi-
ciently extensive experimental data consisting of concrete narrative texts like
annals or chronicles. It turns out that the suggested models can be confirmed.
xiii
XIV
Preface
In other words, we managed to discover interesting statistical regularities con-
trolling the chroniclers’ process of creating long narrative texts. The discovery
of these laws is one of the principal results of our work. And on their basis, the
methods for dating the events described are offered, for which the texts under
investigation are statistically compared with those whose dating is undis-
puted. The methods are then verified against sufficiently extensive concrete
material. We see that their application to texts describing the events from the
13th to the 20th century supports the efficiency of the method. Namely, the
statistical datings obtained are consistent with those that had been known
previously and were established by traditional methods. In particular, textual
pairs originating from common primary sources, and known a priori as depen-
dent between the 13th and the 20th century, also turn out to be dependent
from the point of view of our methods; and pairs of texts known as positively
independent prove to be independent from the standpoint of our methods as
well.
The discovery of the laws that govern the distribution of information in large
historical texts , with the establishment and experimental verification (based on
these laws) of new dating methods (there being eight of them at present ), is
the first basic result of our work . Certainly, the dates we obtained cannot
be regarded as absolute and final. Therefore, we will speak in the following
only of “statistical datings”, although, for brevity, we will sometimes omit the
term “statistical”, which is always implied. We thereby regard the obtained
empirico-statistical dates only as a formal result of the statistical experiments
carried out with narrative texts and do not believe that they are undisputed.
Meanwhile, the consistency of these dates with those known earlier and ob-
tained by the classical methods points to the objective character of our results.
The second fibre. This fibre can be called “critical”. Here, we analyze the
traditional datings of events of the ancient and medieval history of Europe,
Egypt, and the Mediterranean. To make it convenient for the reader, we gather
here the vast data scattered throughout the scientific literature, known to
the specialists of various disciplines (however, often not of general common
knowledge), and shall reveal the serious difficulties on the way to justifying
the dates of certain ancient events.
We shall inform the reader of the fundamental research of the remarkable
Russian scientist and universal scholar N.A. Morozov (1854-1946), Honorary
Member of the USSR Academy of Science, who was the first to pose and fully
formulate the problem of justifying ancient chronology by means of the meth-
ods of natural science, and who collected enormous critical material, putting
forward daring hypotheses. We also speak of Isaac Newton’s chronological
research (questioning the dates of many ancient events), of well-known repre-
sentatives of the critical school in chronology, and of various others working
in the field. We then let major specialists in archaeology, source research, or
numismatics speak and often resort to quoting and supplying the opinions of
well-known scientists, juxtaposing different points of view so that the reader
Preface
xv
can form his own attitude toward the problems touched upon. The analysis of
the dates of ancient events is the basic application of the empirico-statistical
methods we worked out. I was therefore forced to analyze possibly all pre-
served versions of the datings of particular events. As a matter of fact, ancient
and medieval texts often differ with respect to the dates of many important
events. Attempting to stay as close as possible to the “original” versions (and
perhaps to reconstruct them), we usually preferred the versions established in
the chronological documents from the 11th to the 16th century. The chronol-
ogists of that time were nearer to the ancient events described, which is very
important. The versions recorded between the 17th and the 20th century are
often the consequence of later, secondary treatment, sometimes blurring the
original chronological scheme. The reader should always remember this when
looking at the dates given in this book.
Let me clarify this thought. Consider the evolution in time of historical
documents and that of attitudes toward the datings of the described events.
In the absence of a unique system for denoting dates in antiquity and the
Middle Ages, the same events and documents could be dated differently by
different chronologists belonging to different epochs. Let an event occur in
the year to and be fixed in a document X written in to (or around this time)
by a contemporary. X starts “living” when generations succeed each other.
Another chronicler living in a later year t could no longer have access to all the
necessary information and might “calculate” the date of an event. Denote by
D(t o, *) the date ascribed to an event in X, and actually occurring in *o, by a
chronicler who lived in the year f. It is clear that D(to,t) can be different from
to by some positive or negative value. The chronicler’s version of the date can
turn out to be older (then D(to,t) is less than to) or, on the contrary, younger
(then D(to } t) is greater than t). Thus, D(to,t) establishes the point of view
elaborated by the chronicler in t with respect to the datings pertaining to X .
It is obvious that D(to } t) is dependent either of *o or t. We can assume that
D(to,to) = to> ic., the contemporaries mostly date the contemporary events
correctly.
Let us construct the graph of the dependence of D(t 0 i t) on t for a fixed
<o- We then obtain the visual representation of the evolution of the later
chroniclers’ view of dating an event actually occurring in to . It is convenient
to represent it in the form shown in Fig. A. Dating the event t 0 , given by
the contemporary chronologists, is denoted by £>(*0,1986). In other words,
D(*o, 1986) indicates the modern version of the dating if the event actually
occurred in *o. Of course, £>(*o, 1986) can be different from the true dating,
for example, be more ancient or younger.
Since the same event in *o could be described by the contemporaries in
several different documents (this being the typical situation), these individual
versions start existing individually as separate texts not related to each other
from the viewpoint of subsequent generations. We have represented this fact
schematically in Fig. B by doubling or repeating some events and their datings
several times. The further evolution of each version is represented by its own
XVI
Preface
Figure A. Visual representation of the later chroniclers f view of the evolution
of dating an ancient event .
curve, each emanating from the same diagonal but subsequently behaving
absolutely independently. Meanwhile, different versions of the description of
an event, outwardly totally different, can diverge far from each other from
the standpoint of later chronologists. The complete evolution of datings of
ancient events is given in Fig. C. Each subsequent epoch finds its own attitude
toward the datings of the past events. These versions can vary substantially
with time (we give examples of this in the book). Starting with the period
from the 16th to the 17th century (see Fig. C), the chronological version of
ancient times suggested by I. Scaliger and D. Petavius, is being “stabilized” ,
Figure B. Different versions of the description of an event can diverge far
from each other from the standpoint of later chronologists.
Preface
XVII
Figure C. The complete evolution of dating ancient events.
and the modern point of view coincides, in its basic features, with their
chronology. This circumstance shows in the increasing straightening of the
“alignment” of the dating trajectories. Today, we have assimilated only this
version. However, very little is known about all the previous versions, which
often differ sharply from today’s. In other words, we are only well aware of
the topmost line for the dates D(to i 1986) and know very little of other lines,
which obviously make up the bulk of the diagram. Thus, the enormous base
of the chronological iceberg is hidden, within which the modern version of
ancient chronology has been formed. The basic question formulated in the
critical fibre of the book (the second “fibre”) is related only to the under-
water part. It is in this sense that we paid so much attention to the ancient
chronological versions of the 10th- to 15th-century scientists.
We now elucidate what is meant by “correct” , or “authentic” , chronology
in terms of the graphs in Figs. A, B, and C. It is the chronology in which the
evolution of the date of an ancient event would be represented by approxi-
mately vertical lines (see Fig. D). Only in this case can the dates accepted
today be regarded as realistic. To verify whether today’s ancient chronol-
ogy satisfies this condition, we should exhibit chronological tables associated
with each horizontal line in Fig. C, made up by subsequent chronologists of
ancient times. In other words, we should find the originals of those ancient
xviii
Preface
Figure D . Correct or authentic ancient chronology . The evolution of the dates
of events is represented by vertical lines.
chronological versions forming the steps of a staircase which the dates were
“ascending” . Meanwhile, we have to see that the transition from each version
to the previous or subsequent one is represented by vertical lines in Fig. C.
However, an attempt to descend into the past on these “steps”, say by
jumping over the 20 to 30 years that make up a generation, permits us to
move only to the 12th and 13th centuries (with the “staircase” breaking earlier
than that). Here we only discover “dating” of pieces that are not united into
chronological tables preserved until today, and which fix the viewpoints of
the ancient chronologists. Earlier than approximately the 13th century, no
sequence of “shorter” predecessors of the chronological table can be found. It
is desirable that the “shortening” of the table (respectively, its “extension”)
occurred approximately by 20 to 30 years, in the hope that the events of
this time were described by a contemporary. The important characteristic
of the second “fibre” is that the critical material is gathered only here. It
thus acquires a new quality and permits us to embrace a larger volume of
critical data on the basis of one point of view, accumulated in special works
on ancient chronology. We assume that the reader is at least roughly familiar
with tradtional ancient chronology (having studied it at school, university,
Preface
xix
etc.) In general, we do not repeat the traditional version, since we believe it
to be known by everyone, but rather focus our attention on the account and
criticisms of the competing versions, which are sometimes much different from
the traditional one and were developed by many scientists between the 16th
and the 20th century.
Within the framework of the second “fibre*’, we also supply a brief analysis
of the traditional dating methods based on archaeological or radiocarbon data,
which is of use if the reader would like to estimate the degree of reliability
and accuracy. We shall also pay much attention to the dating of events that
are about one, two, or three thousand years old and will demontrate, agai^y^
by citing a number of authors, the difficulties that arise. Dating of material
more than three thousand years old is beyond the scope of this book.
The third fibre. The author has constructed the so-called global chronologi-
cal diagram (GCD), which can be regarded as a sufficiently complete and tra-
ditional “textbook” for ancient and medieval chronology. All the basic events
of ancient history with their traditional dates, lists of the names of principal
characters, and so forth, have been plotted on the time axis, and the basic
preserved primary sources marked for each epoch. The diagram contains tens
of thousands of dates, names, references. Occupying an area of several tens of
square metres, it is a convenient collection of statistical data and a guidebook
to the building of the traditional version. The graphic representation along the
time axis of the principal dates proved useful for the statistical experiments.
Since the GCD contains too much material, it was included in this book only
in abbreviated form as short tables or graphs and is often replaced by this
shorter version. We stress once again that the GCD is based on the traditional
dating of ancient events, arising from the Scaliger and Petavius chronology.
The fourth fibre. The whole set of empirico-statistical methods we devel-
oped was applied to the GCD statistical material (see the first “fibre”). All
possible pairs of time intervals (epochs) along with the basic texts describing
them were considered, and the texts were statistically examined and com-
pared. The “proximity coefficients” or textual “dependence coefficients” were
subsequently calculated. If the dependence coefficient for two texts X and Y
was the same (in order) as for two a priori, positively dependent texts from
the 13th to the 20th century, then X and Y along with the associated time
intervals were called “statistically dependent”. This was represented in the
GCD by denoting the corresponding time segments by the same symbols, for
example, by the same letter T. The symbols were chosen arbitrarily. How-
ever, if the proximity coefficient was the same (in order) as for two a priori
independent texts from the 13th to the 20th century, then X and Y were
termed “statistically independent” and hence represented by noncoincident
symbols like the letters H and C. We would like to make it clear that by in-
vestigating experimentally reliably dated texts describing the 13th to the 20th
century, it was discovered that the proximity coefficients distinguish between
XX
Preface
a priori dependent and independent texts. For example, one of these coeffi-
cients, p(X y Y), did not exceed 10“ 8 for two texts known previously as depen-
dent and was not less than 10~ 3 for two surely independent texts, which shows
the difference of about 4-5 orders. Now, comparing two arbitrary texts X and
y, we can say whether the value of the coefficient is in the zone for dependent
or independent texts. It can also be in the zone of “neutral” texts. It goes
without saying that the indicated bounds for the values of the coefficient have
been found experimentally. Further discovery of dependent and independent
texts is then carried out within the framework of the experimental material
(which is, though, sufficiently large).
For vast computational experiments, the GCD revealed pairs of statisti-
cally dependent texts and the corresponding epochs. The results of applying
different methods turned out (and this is very important) to be remarkably
consistent; namely, if a pair of texts (and periods) were statistically depen-
dent from the standpoint of one method, then they were also dependent from
the point of view of other methods applicable, in principle, to the tests in
question. This consistency seems to be important. Our methods discovered
no unexpected, formerly unknown duplicates of documents belonging to the
period from the 13th to the 20th century. However, for documents preceding
the 13th, and especially the 10th century, the same methods led to the quite
unexpected discovery of many new statistical duplicates regarded as indepen-
dent in all respects, and referring to different epochs.
The global chronological diagram showing all statistical duplicates is the
second principal empirico- statistical result obtained by the author.
The third basic result is the decomposition of the GCD into the sum of four
chronicles practically identical to each other, but shifted by considerable time
intervals. To give a rough idea, the third statistical result can be formulated in
the following way: The modern “textbook” of traditional ancient and medieval
chronology and history is the sum, from the statistical point of view, of the
four replicas of one shorter “chronicle” .
The principal part of the book concentrates on these three empirico-statisti-
cal results. The subsequent “fibres” are mostly of hypothetical and interpreta-
tional character. Roughly speaking, they are required so that we may answer
the question: What do the obtained empirico-statistical results mean?
The fifth fibre. It can be called interpretational. Here, we offer different hy-
potheses which can explain the regularities discovered and the reasons for the
duplicate appearance. We do not regard this material as final. The “shorter”
textbook I suggested certainly does not claim completeness and can only be
regarded as a possible version. Interpretations of the obtained statistical re-
sults can be of different nature and will require much work by many specialists
in various fields.
My attitude toward many of the questions discussed is a result of coop-
eration and numerous discussions. In particular, the statistical results were
reported at the Third, Fourth, and Fifth International Vilnius Conferences
Preface
XXI
on Probability Theory and Mathematical Statistics in 1981, 1985, and 1989,
respectively; the First World Congress of the International Bernoulli Soci-
ety for Mathematical Statistics and Probability Theory in 1986 in Tashkent;
the seminar “Multidimensional Statistical Analysis and Probabilistic Mod-
elling of Real-Time Processes” by Prof. S.A. Aivazyan (Central Economical-
Mathematical Institute, Moscow); the All-Union Seminar on the Stochas-
tic Continuity Model and Stability Problems by Prof. V.V. Kalashnikov
(All-Union Systems Research Institute) and Prof. V.M. Zolotaryov (USSR
Academy of Science. V.A. Steklov Mathematics Institute); Controllable Pro-
cesses and Martingales by Prof. A.N. Shiryayev (USSR Academy of Science.
V.A. Steklov Mathematics Institute).
I would like to express my indebtedness to the participants in the discus-
sions.
I am also indebted to Acad. Ye.P. Velikhov and Acad. Yu.V. Prohorov for
their assistance.
My work received great stimulus from numerous private talks, consultations,
and discussions with colleagues, and also from specialists in mechanics and
from physicists; in particular, I am much indebted to the Moscow University
staff members Prof. V.V. Kozlov, Prof. N.V. Krylov, Prof. M.M. Postnikov,
Prof. A.S. Mishchenko, Prof. Ye.M. Nikishin, Prof. V.A. Uspensky, Prof. P.L.
Ulyanov (Assoc. Member of the Academy of Science), Prof. Ye.V. Chepurin,
Prof. Ye.G. Sklyarenko, Prof. V.I. Piterbarg, Prof. V.V. Moshchalkov, Prof.
M. K. Potapov, Prof. N.V. Brandt, Prof. R.N. Kuzmin, Prof. V.V. Surikov,
Prof. Yu.P. Gaidukov, Prof. Yu.P. Solovyov, Prof. Ya.V. Tatarinov, Prof.
V.V. Alexandrov, Cand. Sci. N.N. Kolesnikov, Cand. Sci. G.V. Nosovsky,
Prof. V.M. Zolotaryov and Prof. A.N. Shiryaryev (USSR Academy of Sci-
ence. V.A. Steklov Mathematics Institute), Prof. V.V. Kalashnikov and Prof.
V.V. Fyodorov (USSR Academy of Science. Systems Research Institute), Prof.
S.T. Rachev (Santa Barbara, USA), and D.I. Krystev (Mathematics Institute;
Sofia, Bulgaria), Prof. Yu.M. Kabanov (USSR Academy of Science. Central
Economical-Mathematical Institute), Prof. A.V. Chernavsky (All-Union Re-
search Information Transmission Problems Institute), Cand. Sci. I.A. Volodin
(Moscow Oil and Gas Institute), Prof. S.V. Matveyev (Chelyabinsk Univer-
sity), and Cand. Sci. M.V. Mikhalevich (Kiev University), Prof. Yu.M. Lotman
(Tartu University), Prof. V.K. Abalakin (Leningrad), Prof. M.I. Grossman
(Moscow), Prof. L.D. Meshalkin (Moscow), Prof. R.L. Dobrishin (Moscow),
Prof. I.Z. Schwartz (Moscow), and Cand. Sci. S.Yu. Zholkov (Moscow), and
Cand. Sci. L.E. Morozova (USSR Academy of Science. History Institute).
My special thanks go to my colleagues Prof. V.V. Kalashnikov and Cand.
Sci. G.V. Nosovsky for their support and collaboration on many problems of
mathematical statistics, astronomy, and computer experiments.
I would like to express my gratitude to all of them.
In addition, I would like to thank Cand. Sci. G.V. Nosovsky, Cand. Sci.
N. S. Kellin, P.A. Puchkov, M. Zamaletdinov, A.A. Makarov, N.G. Chebo-
taryov, Ye.T. Kuzmenko, V.V. Byasha, T. Turova, L.S. Polyakova, my parents
XXII
Preface
V.P. Fomenko and Cand. Sci. T.G. Fomenko, and my wife Cand. Sci. T.N.
Fomenko for their help with the primary statistical processing of the historical
sources, statistical tables, and frequency graphs.
Much help in computer programming and processing of the statistical ma-
terial was given by Cand. Sci. G.V, Nosovsky, Cand. Sci, N.S. Kellin, Cand.
Sci. N.Ya. Rives, Cand. Sci. I.S. Shiganov, P.A. Puchkov, M. Zamaletdinov,
and A.V. Kolbasov.
I would like to express my debt to T.G. Zaharova, the Director of the N.A.
Morozov Museum, and V.B. Biryukov for their assistance in studying the
material related to Morozov’s scientific work.
Further, I would like to thank Kluwer Academic Publishers and, in particu-
lar, Prof. M. Hazewinkel and Dr. David J. Larner for their effort in publishing
the book, with special thanks to Dr. Larner, the head of the Science and
Technology Division, for his support of this project.
I am also grateful to Mr. V.V. Novoseltsev of the Copyright Agency of the
USSR for his help.
And, finally, this book could never have been published without the inter-
ested attention and initiative of the distinguished mathematician and editor
Prof. M. Hazewinkel, who made it possible to combine and publish an En-
glish translation of all my basic works on this problem, which had already
been published in the USSR as individual articles.
The book is dedicated to the memory of the outstanding scientist and uni-
versal scholar Nikolai Alexandrovich Morozov (1854-1946), Honorary Member
of the USSR Academy of Science, author of many profound works in chem-
istry, physics, mathematics, astronomy, and history. It was he who first posed
the problem of scientifically substantiating ancient chronology by using the
methods of natural science, and who obtained fundamental results.
In conclusion, I would like to emphasise that, fully aware of the unusualness
and unorthodox nature of certain of the obtained results, I nevertheless believe
it my scientific duty to present the work to the reader’s judgement in hope
that it may serve as the next step in working out new statistical methods for
the study of narrative sources and in solving the problem of justifying ancient
chronological dates.
The book contains only part of the obtained results. I hope to publish others
separately. In particular, the following books have recently been published:
A.T. Fomenko, Methods for Statistical Analysis of Narrative Texts and Ap-
plications to Chronology , Moscow University Press, 1990 (in Russian).
A.T. Fomenko, V.V. Kalashnikov, and G.V. Nosovsky, Geometrical and Sta-
tistical Methods for Dating Ancient Star Catalogues (When Was Ptolemy’s
“ Almagest ” Compiled in Reality?), (in Russian; English translation in prepa-
ration).
CHAPTER 1
Problems of Ancient and
Medieval Chronology
§1. The Global Chronological Diagram of Ancient and Medieval
History 1
1.1. The moon’s elongation and R. Newton’s copjecture
Chronology informs us of how much time has passed since a certain historical
fact. Meanwhile, the chronological data of a narrative source describing the
fact should be reduced to the modern dating units, i.e., be referred to by B.c.
or a.d. This problem proves to be quite complicated, since many a historical
inference depends on which date we ascribe to the events discussed in the
source.
Modern global chronology embracing the majority of events of the past is
the result of the lengthy work of chronologists who lived from the 15th to the
19th century A.D. Thus, all the major events of ancient and medieval history
are associated with certain dates in the Julian calendar, which permits us
to study historical processes, evolution of scientific and cultural ideas, tech-
nological progress, and so forth, within the scope of large time intervals [1],
[2], [4].
However, such research has led to the discovery of certain phenomena which
cannot easily be explained for the present. We give an example from natural
science, namely, from astronomy. The lunar theory deals, inter a/ia, with a
parameter called the second derivative of the moon’s elongation (D"). De-
pending on time, the values of this parameter should be available for past
eras. It can be computed if the ancient eclipse data are known. The problem
has been solved by the prominent American astronomer E. Newton [10]. The
1 First published as an article in Kkimiya t zkyzn\ 9(1983), pp. 85-92.
1
2
Problems of Ancient and Medieval Chronology
Chapter 1
Figure 1. R. Newton’s graph demonstrating that D n (i) decreases with time.
See the astonishing, inexplicable jump at around the first millenium A.D.
graph (Fig. 1) he obtained turned out to be extremely surprising. Newton
wrote:
“The most striking feature of Fig. 1 is the rapid decline in D n from about A.D.
700 to about A .D. 1300. . . . This decline means that there was a ‘square wave* in the
osculating value of D n . . . . Such changes in D", and such values, are unexplainable
by present geophysical theories. ...” ([10], p. 114)
To explain this square wave (one-order jump), Newton was forced to suggest
that there should exist some nongravitational interactions in the earth-moon
system [11]. These enigmatic forces do not manifest themselves in any other
way, which is in itself quite unusual.
Below, we shall see that there is at least one more explanation of the jump
in D".
1.2. The Dark Ages and the Renaissance epochs
Let us return to chronology. In the history of Europe and the Mediterranean,
there are several Renaissance epochs during which many achievements of an-
cient scientific thought, lost in the period of the Dark Ages, were discovered.
The epoch in the history of Europe when many scientific facts and cultural
habits of the past were rediscovered (from the 13th to the 16th century) has
been studied most extensively. Such duplication is explicitly traced in astron-
omy, military engineering, architecture, literature, and many other branches
of science and art. For example, the famous Greek fire, which had played such
an important role in the sea battles of antiquity, and which had then been
forgotten for centuries, was rediscovered only in the Middle Ages.
§1
The Global Chronological Diagram
3
Apart from the classical Renaissance, the Carolingian Renaissance (the time
of Charlemagne) is also generally known, when many authors imitated the an-
tique paragons, duplicating the literary themes which had been forgotten ear-
lier. Similar phenomena (termed Restoration) are also known in the history of
ancient Egypt. The prominent Orientalist B.A. TViraev noted that the culture
of the Saite period had reproduced that of the Old Kingdom: 2,000-year-old
texts again went into use, tombs were decorated following the ancient ways,
titles that had sunk into oblivion were reintroduced, and so forth.
As we see, duplicates present themselves as a rather frequent phenomenon
in history. Naturally, the question arises as to how they are distributed in
time: in a random manner or subject to some covert governing law?
1.3. How to substantiate ancient chronology
To calculate the dates of ancient events is not as simple as it may seem at
first glance. The final proof of the correctness of certain dates still remains
a problem today. It continues to attract the attention of historians and the
specialists of physical and chemical dating methods. It is but natural: The
further we move from an ancient event in time, the harder it is to date it.
The contradictions that often arise in doing so have caused some historians to
express doubts regarding the dating of certain events, as suggested by the first
chronologists of the 16th to the 18th century, which, by the way, are still ac-
cepted with few exceptions at present [12], [13]. A new scientific discipline was
born, namely, hypercriticism, which denied not only the correctness of dating
a particular event, but also the trustworthiness of certain ancient events. The
famous representative of this school, who specialized in the history of ancient
Rome, T. Mommsen, noted, in particular, that different versions of dating
the foundation of Rome diverged to the extent of 500 years, and that this
oscillation influenced the dating of all the documentation counting years since
the “foundation of Rome” ([14]; [14*], pp. 513-514).
Chronological problems interested the Egyptologists, too. Thus, H. Brugsch
stressed the enormous difference in the determination of the date when Menes
had been placed on the throne, writing that the difference between the extreme
conclusions was striking, it being equal to 2,079 years. In spite of all the
discoveries in this branch of Egyptology, the numerical data were (at the end
of the 19th century) still in a very unsatisfactory state ([16]; [16*], pp. 95-97).
Another example: The chronology of cert sun events in Egyptian history,
which was given by Herodotus in his famous Histories , differs by more than a
millennium from that accepted today. Herodotus’ chronology is much shorter
than the modern version; sometimes, he even places near each other (see [17])
rulers who according to the modern version are separated by 18 centuries
([17*], pp. 512, 513, 516).
But especially many discrepancies show up if one compares the dates given
in medieval texts with the dating ascribed to them today. The distinguished
4 Problems of Ancient and Medieval Chronology Chapter 1
modern chronologist E. Bickerman even speaks of “the chaos of medieval
datings” ([4], p. 78).
Chronology in its present form was created in a series of fundamental works
by the founders of modern chronology as a science, J. Scaliger (1540-1609) and
D. Petavius (1583-1652). It became a precise science later; however, the work
is not yet completed, and, as Bickerman notes, there is no sufficiently complete
investigation of ancient chronology that would satisfy modern requirements
(M; [ 4 *]> p- 90).
It is not surprising that certain sceptical minds have drawn dramatic conclu-
sions from the above-mentioned difficulties. Thus, as early as the 16th century,
a professor of Salamanca University, de Arciila, published two papers in which
he stated that the whole of history preceding the 4th century had been fal-
sified (see de Arciila, Programma Historicae Universalis , Divinae Florae His -
toricae). The same conclusion was reached by the historian and archaeologist
J. Hardouin (1646-1724), who regarded the entire classical literature as the
work of 16th-century monks. Isaac Newton devoted many years to historical
and chronological studies. Having thoroughly investigated practically the entire
historical and theological literature, he wrote Abregis de la Chronologic [19],
asserting that the time scale of the chronology of antiquity was unnaturally
extended. Newton made up his own tables in accordance with a new version of
chronology which related the biblical texts to the history of the Mediterranean.
In his book Newton , V.G. Kuznetsov wrote that Newton had collected
“fantastically large volumes of historical material. This was the total of forty years
of work, toilsome research and enormous erudition. Newton, in fact, studied all the
basic literature in ancient history and all primary sources . . . ” ([18], pp. 104-105).
“Certainly, being unable to read cuneiform and hieroglyphic texts and having no
archaeological data, which were then unavailable, . . . Newton was in error to the extent
not only of tens or hundreds of years, but even millennia ...” ([18], pp. 106-107).
As a matter of fact, many of the most important events of Greek history
were chronologically moved forwards by Newton by 300 years, and those of
Egyptian history by 1,000 and even 1,800 years.
And now in this century, in his Historic und Kritik , the German researcher
R. Baldauf was proving on the basis^of philological arguments that not only
ancient but even early medieval history was a later falsification.
An attempt to systematize the considerable critical material and to analyze
historical paradoxes and duplicates from the standpoint of natural science was
carried out in the work of a scientist with encyclopaedic knowledge, the revo-
lutionary, public figure and honorary academician, N.A. Morozov (1854-1946)
[3]. He actually held the opinion of de Arciila and believed that traditional
chronology had been artificially stretched [2]. It should be noted that he ap-
parently came to this idea independently of de Arciila. Remarkable scientific
intuition and strict logical argumentation permitted Morozov to list numerous
data in support of such a conjecture. However, his striving to dot all the i’s
led to poor substantiation of many of his statements; some contained factual
§1 The Global Chronological Diagram 5
errors, and the new chronological version as a whole (including the hypothesis
regarding the falsification of ancient history) was rejected, which does not at
all lessen his achievements, for the problem is so complicated and many-sided
that one mind alone, even if outstanding, is unable to solve it completely.
1.4. Statistical dating methods: new possibilities
To overcome the above difficulties, we should try to consider the subject from
a different angle and create a certain independent dating method which is not
based on subjective estimation. This done, we can start analyzing the whole
of chronology. In my opinion, an approach involving the statistical analysis of
various numerical characteristics associated with ancient texts is most suitable
for this purpose. The interested reader can learn about concrete methods and
some of their applications to the analysis of global chronology from the short
bibliography at the end of the section. Here, we shall confine ourselves to a
short account of the essentials and give several examples.
We make the immediate reservation that the methods suggested by the
author do not pretend to be universal. Moreover, the results obtained by
each individual method cannot be regarded as impeccably trustworthy. A
sound criterion of their validity is the consistency of the dating obtained by
different methods (today, there are seven of them). The general scheme is
as follows. First, a statistical hypothesis is formulated for modelling some
process (e.g., loss of information with time). Then numerical coefficients are
introduced which permit us to quantitatively measure the deviations of ex-
perimental curves from those predicted theoretically. Further, the model is
checked against a priori true historical material, and if it is confirmed, then
the method can be used for the dating of events.
For simplicity, we give an example. Let a period in the history of a region
P from a year M to a year N be described in a text X (chronicle or annals)
broken into separate chapters X(T ), each of which is devoted to the events
of a year T. We calculate the volume of all the chapters (number of pages or
lines) and represent the obtained data as a volume graph, plotting the years,
T, on the horizontal, and the volumes of the chapters on the vertical axis.
A similar graph for another text, Y, describing the same events, in general
will have different form; most probably, the interests or tendencies of the
chronologist will have bearing on it. But how essential are these differences?
Is there anything common between the volume graphs? Indeed, there is. But,
before stating the details, some words about the information- loss pattern.
The essential characteristic of any graph is its peaks, or extremal points.
In our volume graph, they correspond to the years in which the curve attains
local maxima. Such peaks indicate the years described by the chronicle in the
time interval under investigation with the finest points of detail. Denote by
C(T) the volume of all texts created by contemporary writers and describing
a year T. We call it the “primary stock” (Fig. 2a). The precise form of its
graph is not known to us, since texts get partly lost in the course of time.
6
Problems of Ancient and Medieval Chronology
Chapter 1
Primary information Surviving stock
Dependent Textual Textual
Figure 2. Textual volume graphs in a time interval MN: (a) the primary
and surviving stocks (curves should exhibit peaks approximately in the same
years); (b) the curves for dependent texts are correlated; (c) the curves for
independent texts are not correlated.
We now formulate the information-loss model, namely, there will be more
texts for those years to which more texts were originally devoted. It goes
without saying that to verify the model in this form is difficult, because the
graph of the primary stock remains unknown. But one of the corollaries can
§1
The Global Chronological Diagram
7
be verified. Later authors, X and Y , while describing the same period (and
not being its contemporaries), will be forced to employ approximately the
same set of ancient texts available. Therefore, they will be able to describe
best those years from which more texts remain.
Eventually, the model conjecture is formulated as follows: The graphs of
the volumes of chapters for two dependent texts X and Y (i.e., describing the
same period of history and the same region) must have simultaneous peaks ;
in other words, the years described in detail in X and Y should coincide or
be close (Fig. 2b). On the other hand, if two texts X and Y are independent
(either describing essentially different periods of history of the same length,
or for different regions), then the graphs of the volume for X and Y attain
local maxima at different points (Fig. 2c).
After the mathematical formalization, an experiment was carried out in
which the model ( maximum correlation principle) was verified for several hun-
dreds of pairs of dependent and independent historical sources. The principle
was confirmed, which made it possible to offer a method for dating texts and
also for discovering interdependency among them. For example, to date events
described in a chronicle, we have to try to choose an a priori dated text such
that the volume graphs attain maxima practically simultaneously. If, however,
the dating of two comparable texts is unknown, but the peaks in the graphs
coincide, then we can assume with a high degree of probability that the texts
are dependent, i.e., the events described are close or coincide.
Now, just a few words about some other methods of dating. They are based
on the statistical analysis of such parameters as the frequency of mentioning
the names of historical characters and that of various astronomical phenom-
ena, the period of the rule of kings in various dynasties, formalized biograph-
ical data of historical figures, and so on. All these methods have been verified
for undoubtedly true material of the 13th to the 20th century, and their va-
lidity has been confirmed [5], [6], [7],
1.5. The duplication effect in ancient history and chronology
The methods briefly described above are applicable not only to the dating
of texts. They also permit us to find various literary borrowings, repetitions,
literary cliches, citations, and parallels in the texts being compared. For exam-
ple, if in comparing two dynasties of kings a certain dependence is discovered
(i.e., if the corresponding graphs of the rule duration are extremely close),
then this can be interpreted in different ways. One interpretation consists
of our probably having discovered an intentional imitation by the annalist
of a certain authoritative source. However, another version is also probable,
namely, that we are dealing with duplicates which were never recognized to
be identical and are narrating the same events, and which were related to
different historical epochs.
Sometimes these methods help discover the proximity of chronicles, i.e.,
their having originated from the same source. In particular, they make it
8
Problems of Ancient and Medieval Chronology
Chapter 1
possible to indicate the duplication effects discussed at the beginning of the
section in connection with the Renaissance epochs. As it turns out, there are
essentially more such historical epochs than is usually thought. To avoid a
terminological muddle, we will speak in the following simply of duplicates.
It is now time to formulate our problem: to find possibly all duplicates in
ancient and medieval history, and if we succeed, to construct on their basis
a hypothetical chronology without repetitions and Renaissance periods which
are sometimes hard to explain.
1.6. The global chronological diagram and the “modern
textbook” of ancient and medieval history
Before coming to the thorough analysis of historical texts for the purpose of
discovering and systematizing duplicates, we have to construct as complete a
table of events of the ancient and medieval history of Europe as possible and
also that of the Mediterranean region, Egypt, and certainly the Near East,
showing their traditional dating. To this end, the author has investigated 15
basic chronological tables and 228 fundamental primary sources (chronicles,
annals, records, etc.). Together, these texts contain the description of practi-
cally all the basic events of the period from 4000 B.c. to a.d. 1800. All this
information was then represented graphically on the plane. Each historical
epoch with all of its basic events was shown on the time axis. Meanwhile,
each event was represented by a point or horizontal line-segment in accor-
dance with its duration, with the beginning and end of the line-segment
being those of the event (e.g., a king’s rule). Simultaneous events were rep-
resented one above the other, so that any ambiguity or overlapping would
be avoided.
Thus, a maximally complete chart was constructed. We will call it the global
chronological diagram (GCD); see Fig. 3, upper line. To see which events took
place in a particular year according to modern chronology, we have to draw
a vertical line through that year on the GCD and collect all the events being
intersected.
We have applied the above dating and duplicate-recognition methods to the
enormous historical data recorded on the GCD. The entire period of history
on the diagram was broken into epochs, for which, roughly speaking, a set
of characteristic graphs was calculated. For example, for each epoch (a line-
segment on the time axis) in the history of each region, the volume graphs
for all the basic primary sources describing this epoch were plotted, and those
for different epochs were compared pairwise. As a result of the extensive ex-
periment in which hundreds of texts were investigated, containing altogether
tens of thousands of names and hundreds of thousands of lines, the author
unexpectedly discovered pairs of epochs which are regarded as independent in
traditional history (in every sense), but with extraordinarily close and some-
times even practically indistinguishable graphs.
§1
The Global Chronological Diagram
9
-1700
-1500
i
-1100
-900
-700
-500
•300
8
S i
?
1 1 § 8 §
1
European history (chronicle E)
1 1
1 1
7
T T T
T
T T
mi
auE
T T
B
!W
H
m
1 1
!□!<
w
fm
B
J
MM
□1
IB
■1
ZJI
c
FI
C
1
T T T
T
1 p l
C 4 :
e
m
m
1.778-year shift
T
T
T
C 3 :
1
1 .053-year shift
T
m
T
(
* ,
J 1‘
liar
ID
11 C 1
333-year shift
Ini
lul
T
r
C,:
y
nor
ITT
nr
Figure 3. The “modem textbook * of European history and its decomposition
into the sum of four short isomorphic chronicles.
We illustrate this with an example. The volume graph of the primary sources
describing the history of ancient Rome from 753 to 236 B.c. exhibits peaks
practically in the same years as a similar graph constructed for medieval
Rome from A.D. 300 to 816. To verify this fact, these two time intervals of 500
years in length should be superimposed first (Fig. 4). The same coincidence
of the two seemingly independent series of events (antique and medieval) was
also discovered by other methods. The GCD happened to include quite a
number of duplicates, i.e., pairs of historical epochs which are as close as are
undoubtedly dependent texts describing the same historical period. We once
again emphasize that the results obtained by different methods are invariably
consistent.
10
Problems of Ancient and Medieval Chronology
Chapter 1
+300 +758 +816
Figure Primary source volume graph } describing ancient (dotted line) and
latest (solid line) Roman history: this maximum correlation can hardly be
accidental
1.7. The “modern textbook”, a composition of four
identical pieces
Let us once more carefully consider the upper line in Fig. 3. To represent the
set of all the discovered epochs of duplicates clearly, they are marked on the
GCD by the same geometric symbols and letters (chosen arbitrarily). More
precisely, duplicates are designated by the same letters, and the epochs that
are considerably different from one another by different ones.
Some of the letters repeat continually (e.g., T repeats eleven times, and C
four times). The length of the geometric figures indicates the duration of the
corresponding epoch. Say, the black triangles T are associated with periods
that are about 20-30 years long, and the rectangles C with periods approxi-
mately 300 years long. Certain intervals of time on the GCD are covered by
several figures. Thus, the period from ca. a.d. 300 to 550 is represented by
four superimposed rectangles II, A, C, P, which means that part of the chart
devoted to this period is composed of four pieces designated by different let-
ters. In other words, in the set of events which occurred in the interval from
A.D. 300 to 550, those making up the piece II are first distinguished, then
those composing A, and so forth. The events falling into a particular piece
are most often associated by what happened in the region. By the way, all the
Renaissance epochs noted by the historians are contained in the duplicates on
the GCD.
But the main thing is that a rather complicated structure of the GCD is
naturally obtained as the result of one quite surprising process. If the four
lines (chronicles) Ci, C 2 , C 3 , Ca (also shown in Fig. 3) are distinguished in
the chart and are glued together along the vertical line by superimposing, then
we shall obtain, as can be expected, the same line on the GCD, consisting of
§1
The Global Chronological Diagram
11
the lettered epochs. But the most surprising fact is that these four chronicles
are represented by practically the same series of letters and symbols. The four
duplicate pieces differ from one another only by their position on the time axis.
Thus, the second chronicle differs from the first one only by a backward shift
in time of about 333 years, the third by a shift of already 1,053 years, and the
fourth by an approximately 1,778-year-long shift. Admitting a certain liberty,
we can say that the “modern textbook” of the ancient and medieval history of
Europe, the Mediterranean region, Egypt, and the Near East is a composite
chronicle obtained by gluing together four practically identical replicas of the
abridged chronicle C\. Three other chronicles are derived from it by redating
and renaming the events described, while the whole of C\ is lowered (i.e.,
shifted back in time) by about 333, 1,053, and 1,778 years, respectively. Thus,
the entire GCD can be restored from its part C \ .
Another fact to be emphasized is that nearly all the information in the
chronicle C\ is concentrated to the right of A.D. 960. The periods P, T, C (to
the right of the 10th century A.D.) are very rich in information, whereas K ,
H , II (from A.D. 300 to 960) contain very few events.
1.8. Certain corollaries and interpretations
This formal decomposition of the “history textbook” into the sum of four
chronicles can be interpreted differently. First, that the periodic behaviour
I discovered is possibly accidental. It can be calculated, however, that the
probability of such a random event is extremely small. Another possible inter-
pretation is that insufficient written evidence casting light on certain periods
of ancient history encumbers the application of statistical methods. Finally, a
third possible explanation, which seems to me worth notice, is that the exist-
ing global chronology of the period preceding the 13th century A.D. requires
quite substantial corrections in certain cases. These will require the redating of
certain blocks of events now related to earliest antiquity, for which the chron-
icles C 4 , C 3 , C 2 of the modern chronological chart should be distinguished
and lifted upwards in accordance with the mentioned shifts. After this formal
procedure, the known written history of Europe, the Mediterranean, and so
forth, will be abridged, and most of the events now dated as having occurred
earlier than the 10th century A.D. will be placed in the interval from the 10th
to the 17th century A.D.
This hypothesis can help explain certain long-known paradoxes of tradi-
tional chronology, including those mentioned at the beginning of the section.
However, I do not at all agree with the assumption of N.A. Morozov and some
of his predecessors that the information today available regarding ancient his-
tory is, allegedly, a later falsification. The results obtained by new methods of
dating show that most of the primary sources which have been prescribed are
originals describing real events. Almost all the events mentioned in ancient
documents did occur; the question remains only where and when.
12
Problems of Ancient and Medieval Chronology
Chapter 1
Generally speaking, the principal result of the work done is of formal statis-
tical character, and no more. Nonspecialists in history have already attempted
to interpret this result in a pseudoscientific manner, with the data of social
science being ignored. I am decisively against such conclusions.
1.9. What is to be done with the moon 9 s elongation?
Let us return to the beginning of the chapter, to the moon’s elongation and its
second derivative. The computation of D " was based on the data of ancient
eclipses adopted by traditional chronology. The attempts to explain the sur-
prising square wave in the graph of D 11 do not touch at all upon the question
whether the data of the eclipses were determined correctly. We will assume
that an eclipse has been dated correctly if its characteristics exactly described
in a historical source coincide with the parameters of the real eclipse offered
by chronology.
Morozov suggested a method of “impartial” dating, namely, the compari-
son of the characteristics of an eclipse given in a primary source with those
from astronomical tables. Analysis demonstrates that, while not questioning
the chronology of ancient events and a priori regarding it as true, the as-
tronomers often could not find a suitable eclipse in the “desired” century and
thus resorted to strained interpretations. For example, in the History of the
Peloponnesian War by Thucydides, three eclipses were described, tradition-
ally dated as belonging to the 5th century B.C. However, even in the last
century, a discussion around this triad started, being caused by the fact that
there were no eclipses with suitable characteristics in the assumed epoch. Still,
an exact solution can be found if we extend the interval of the search. One
solution is the 12th century A.D., and the second one the 11th century A.D.
There are no other solutions.
A similar effect of “shifting the dates forwards” can be extended to those
eclipses which are traditionally dated in the interval from A.D. 400 to 900. It is
only after A.D. 900 that the traditional dates are satisfactorily consistent with
the precise datings given by astronomy, and undoubtedly after A.D. 1300.
But why, in fact, speak of it here? Because such a shift of dates is completely
consistent with the GCD being glued together from four identical chronicles.
If an earlier and traditional date for an eclipse was assigned to an epoch, say,
labelled by C on the GCD, then its precise astronomical date lies much farther
to the right on the time axis. It occurs in the period of history denoted on
the diagram by the same letter. In particular, the date shift just described is
reduced to advancing certain groups of eclipses up by about 333 years, others
by 1,053 years, and so on. In such a time advance, the mutual occurrence of
dates inside each of these groups is practically unaltered, and the group is
advanced as a block.
But what’s to be done with D"? Its recalculation on the basis of the re-
considered dates of ancient eclipses showed that the graph (Fig. 5) is qualita-
tively altered. It cannot now be moved reliably to the left earlier than the 10th
§1
The Global Chronological Diagram
13
Figure 5. The new graph of D"{t), constructed on the basis of the recalculated
dates of ancient eclipses, has no anomalies: there are simply no reliable data
to extend it to the left.
century a.d., while in the later period, it almost coincides with the curve al-
ready found and is represented by an almost horizontal line. No square wave
is found in the second derivative, and no mysterious nongravitational theories
should be invented. . . .
It goes without saying that the work discussed here cannot claim to be the
basis for any final conclusion, the more so as the most complicated, multi-
farious and often subjectively interpreted historical data are analyzed here
by strictly mathematical methods. To process the material will certainly re-
quire a large variety of methods, purely historical, archaeological, philological,
physical and chemical, and, inter alia , mathematical, which as the reader can
see, will permit us to look at the problems of chronology from a new angle.
References
[1] Blair, J., Blair's Chronological and Historical Tables, from the Creation
to the Present Time, etc. G. Bell & Sons, London, 1882.
[2] Morozov, N.A., Christ. Giz, Moscow-Leningrad, 1926-1932 (in Russian).
[3] Nikolai Aleksandrovich Morozov: A Universal Scholar. Nauka, Moscow,
1982 (in Russian).
[4] Bickerman, E., Chronology of the Ancient World. Thames k Hudson,
London, 1968.
[4*] Russian translation of [4]. Nauka, Moscow, 1975.
14
Problems of Ancient and Medieval Chronology
Chapter 1
[5] Fomenko, A.T., “Informative functions and related statistical regulari-
ties”, Abstracts of the Reports of the Third International Vilnius
Conference on Probability Theory and Mathematical Statistics. Institute
of Mathematics and Cybernetics of AN LSSR 2(1981), pp. 211-212.
[6] Fomenko, A.T., “A method of duplicate recognition and some applica-
tions”. DAN SSSR 258, 6(1981), pp. 1326-1333 (in Russian).
[7] Fomenko, A.T., “New empirico-statistical methods in ordering texts and
applications to dating problems”, DAN SSSR 268, 6(1983), pp. 1322-
1327 (in Russian).
[8] Fomenko, A.T., “The jump of the second derivative of the moon’s elon-
gation”, Celestial Mechanics 25, 1(1981), pp. 33-40.
[9] Newton, I., The Chronology of Ancient Kingdoms amended . To which is
prefix'd, a short chronicle from the first memory of things in Europe , to
the conquest of Persia by Alexander the Great. J. Tonson, etc., London,
1728.
[10] Newton, R., “Two uses of ancient astronomy”, Phil. Trans. Royal Soc .,
Ser. A., 276(1974), pp. 99-116.
[11] Newton, R., “Astronomical evidence concerning non-gravitational forces
in the Earth-Moon system”, Astrophys. Space Sci. 16, 2(1972), pp. 179—
200 .
[12] Scaliger, J., Opus novun. de emendatione temporum. Lutetiae, 1583.
[13] Scaliger, J., Thesaurus temporum: Eusebii ... Chronicorum canonum
omnimodae historiae libri duo ... Opera ac studio J.J. Scaligeri, etc.
Lutetiae, 1606.
[14] Mommsen, T., The History of Rome. Macmillan k Co., London, 1913.
[14*] Russian translation of [14]. Giz, Moscow, 1936.
[15] Petavius, D., Opus de doctrina temporum divisum in partes duas , etc.
Lutetiae Parisiorum, 1627.
[16] Brugsch, H., Egypt Under the Pharaohs. John Murray, London, 1891.
[16*] Russian translation of [16]. Petersburg, 1880.
[17] Herodotus, The Histories of Herodotus , etc. Everyman’s Library, Lon-
don-New York, 1964.
[17*] Russian translation of [17]. Nauka, Leningrad, 1972.
[18] Kuznetsov, V.G., Newton. MysP, Moscow, 1982 (in Russian).
[19] Newton, I., Abreges de la Chronologie des Anciens Royaumes. Chez
Henri-Albert Gosse, Geneve, 1743.
§2
The Moon’s Elongation and Ancient Eclipses
15
§2. Computation of the Second Derivative of the Moon’s Elongation
and Statistical Regularities in the Distribution of the Records
of Ancient Eclipses 1
2.1. Parameter D" and R. Newton’s paper “Astronomical
evidence concerning non- gravitational forces in the
Earth-Moon system”
The present section discusses in more detail my results described in [4] (see
the list of references at the end of §3). It is known that, for certain problems of
computational astronomy, the behaviour of the so-called second derivative of
the moon’s elongation D ,f (t) as a function of time t should have been known
for large time intervals in the past [13]. Let tim be the acceleration of the
moon with respect to ephemeris time, and we that of the earth. The quan-
tity D n = riM — 0. 033862 we, which is the second derivative of the moon’s
elongation, is called an acceleration parameter [10], [13]. D n is normally mea-
sured in arc seconds per century squared. The dependence of the parameter
D ,f (t) on time has been established in a series of remarkable works by the
American astronomer R. Newton [9], [10], [13], who calculated 12 values of
the parameter D " on the basis of the investigation of 370 observations of an-
cient and medieval eclipses, extracted from historical sources ([11], p. 113).
In computing the date < e ci. of the observation of a particular concrete eclipse,
the parameter D u can be neglected. Therefore, it can, in turn, be found from
the distribution of ancient eclipse dates i ec i., which is a priori regarded as
known. In R. Newton’s papers [9], [10], [11], [13], the computation of was
based on the dates of ancient eclipses contained in the chronological canons
of F. Ginzel and T. Oppolzer [8], [12]. They are generally accepted in the
contemporary literature. The results of Newton, related to those of Martin,
who studied about 2,000 telescopic observations of the moon from 1627 to
1860, allowed him to construct an experimental curve for D ;/ (<) in the inter-
val from 900 B.c. to A.D. 1900. In the following, we will sometimes designate
A.D. by “+” , and B.C. by ” . In Fig. 6, the symbol • indicates the values of
the parameter D n calculated by means of solar eclipse data, while 0 denotes
those of D" which were computed from the lunar eclipse durations fixed in
the documents. The sign A implies the values of D" calculated on the basis of
information regarding the duration of solar eclipses. Finally, V indicates the
values of D" computed from the phases of solar eclipses (see [11]).
Commenting upon the graph of D" obtained, Newton wrote:
“D" has had surprisingly large values and ... it has undergone large and sudden
changes within the past 2000 years ...” ([11], pp. 114-115).
1 First published as an article in Operations Research and ACS , Vol. 20, Kiev University
Press, Kiev, 1982, pp. 98-113 (in Russian).
16
Problems of Ancient and Medieval Chronology
Chapter 1
Figure 6. Experimental curve for D ,f (t) in the interval from 900 B.C. to A.D.
1900 (R. Newton).
Newton’s paper “ Astronomical evidence concerning non-gravitation al forces
in the Earth-Moon system” [9] was also devoted to the attempts to explain
this strange gap (one-order jump) in the parameter D n .
Thus, on the basis of Newton’s works [9]-[ll], we can make the following
conclusions.
(1) In the interval A.D. 400-600, the parameter D" starts falling sharply
(one-order jump).
(2) Before this interval, until A.D. 300-400, the values of D n do not deviate
much from zero.
(3) Starting with about A.D. 1000, the values of D n are close to those of
today; in particular, they are practically constant.
(4) In the interval 6000 B.C. to A.D. 1000, the parameter D n undergoes con-
siderable variance, with the oscillation amplitude reaching up to 60" /century 2 .
Hereafter, the bounds of the time intervals indicated are approximate. New-
ton writes that D" “has even changed its sign near about A.D. 800” ([11],
p. 115).
In the following, we shall point out two bounds in the behaviour of the
graph D", the first of them being about A.D. 500 (the beginning of the square
wave on the graph), and the other one about A.D. 1000 (the end of the square
wave).
In the present section, we give the results of a new interpretation and cal-
culations of the graph of D", based on the dates of astronomical observational
data made precise, which form the basis for computing the parameter D u . The
curve of D" which we obtained has qualitatively different character. In partic-
ular, the incomprehensible one-order gap of the graph completely vanishes. As
it turns out, the new graph of D" is, in reality, oscillating around a constant
numerical value which coincides with the modern one. As a corollary, the
§2 The Moon’s Elongation and Ancient Eclipses 17
necessity to invent “nongravitational forces” for the explanation of the “gap”
in the graph becomes unnecessary.
2.2. Available observations of ancient solar and lunar eclipses
Let X be the set of all available observations of ancient solar and lunar eclipses.
Their complete list has been given by F. Ginzel ([8], pp. 167-271). Let A be
the set of all eclipses described in the ancient texts X. We have to bear in
mind that the same eclipse may be described in several ancient texts. We
denote them by Aeci.* Let t ec i be the date ascribed to a particular eclipse
in accordance with the traditional chronology. These traditional dates have
been fixed in the papers of F. Ginzel and T. Oppolzer [8], [12]. They all
form a basis for the computation of D"(t). In computing D'^eci.) (i-e., at
a point t e ci. on the time axis t ), the theoretical, calculated characteristics of
an eclipse, obtained for the date t e c i. on the basis of modern lunar theory,
are compared with the description of the eclipse, portrayed by the ancient
sources X ec i.. The deviation between these two groups of data is exactly what
permits us to find the value of the parameter D n at the moment t ec i. . This
value of D" certainly depends on the choice of the eclipse date, and only those
ancient texts are important which contain sufficiently much information about
it, e.g., the description of the trajectory, phase, and so on. The analysis of all
the ancient texts available (see F. Ginzel [8]) permitted us to distinguish a
list of sufficiently complete descriptions of eclipses. We do not have the space
to give it here. All our computations in the following are related just to these
eclipses.
Newton’s attempts to explain the mysterious square wave of the function
£>"(<) do not touch upon the problem of the precision and correctness of the
dates ascribed to the ancient and medieval eclipses by traditional chronology
[8], [12]. In other words, the question as to how well the descriptive parame-
ters of an eclipse, fixed in an ancient text, correspond to the calculated eclipse
parameters found for the moment on the basis of lunar theory, was ad-
dressed, with i ec i. meaning here the date ascribed by traditional chronology
[8], [12]. The dating and the description of a given eclipse can be regarded as
correct only in the case where the two groups of characteristics, i.e., calculated
and fixed in a historical source, coincide. Note that changing the dates of the
eclipses will alter the graph of D " .
The relation between the problem of calculating the parameter D" and
the known investigations of N.A. Morozov [2] was indicated for the first time
in the author’s paper [4], which, in particular, touched upon the problem of
correctly dating ancient eclipses and their descriptions. On the basis of the
analysis of considerable factual data, Morozov suggested and partly substanti-
ated his fundamental conjecture that the traditional chronology of the ancient
world might be artificially extended in comparison with the real situation. An
important role in forming this conjecture was played by the method of astro-
nomical dating.
18
Problems of Ancient and Medieval Chronology
Chapter 1
The descriptions of eclipses from certain ancient texts started to be em-
ployed for dating these sources and related events as early as the 16th century.
However, the method was applied only for the purpose of obtaining the dates
in a somewhat more precise and, usually, quite narrow, prescribed time in-
terval where traditional chronology placed them and the simultaneous eclipse
under investigation.
2.3. A method of formal astronomical dating
In paper [2], a method of formal astronomical dating was suggested, consisting
of the extraction of the eclipse’s descriptive characteristics from a historical
text, and then purely mechanically recording all dates of the eclipses with
these characteristics from the modern astronomical tables. The recalculation
of the ancient eclipse data was performed just by this method in the indicated
work, with the dates traditionally being ascribed to the time interval from 700
B.c. to A.D. 400. For the purpose of computing the parameter D" , I carried
out a new series of calculations of the ancient and medieval eclipse dates, thus
confirming, in particular, the effect of shifting the dates of ancient eclipses
forwards from 700 B.c. to A.D. 400 ([2]; [4]; see below).
We now describe the method of formal astronomical dating in more detail.
The papers of Ginzel and Oppolzer [8], [12] supply a list of 89 ancient eclipses
and indicate the ancient texts which reported them. The latter are usually
(traditionally) dated to have occurred in the interval from 700 B.c. to A.D.
592. A list of the eclipses’ descriptive characteristics extracted from an an-
cient text can be complete to varying degrees. For example, the moment of
an eclipse during an entire day can be indicated, but not its phase, and so
forth. Besides, the canons of Ginzel and Oppolzer contain the complete and
theoretically calculated list of eclipses occurring from 900 B.c. to A.D. 1582,
with the basic characteristics including the date and phase of an eclipse, the
umbra coordinates, and so on. The problem of dating an eclipse described by
an ancient text (and, therefore, that of the accompanying events) is solved as
follows. We take the eclipses from the canons [8], [12], all of whose calculated
characteristics exactly coincide with those in a historical source. At the same
time, it is required that (1) there should be no deviation from the description
in the document, and (2) the time interval in which the astronomical solution
is sought should not be bounded. This means that we do neglect the a priori
outside information of “nonastronomical origin”.
Analysis shows that requirements (1) and (2) are not fulfilled in the classical
works [8] and [12] in the overwhelming majority of cases: The date of an
eclipse is usually sought there not in the possible whole historical time interval,
but only within narrow prescribed limits (normally, one century) in which,
according to earlier chronological tradition, an approximate date of the event
studied (and, therefore, of the eclipse) was pre-established.
The application of the method of formal astronomical dating to eclipses
§2
The Moon’s Elongation and Ancient Eclipses
19
traditionally dating from 700 B.c. to a.d. 400 [2], [4] shows that the written
evidence concerning them can be separated into two classes.
(1) Short and vague evidence ( without any details ). Here, it is often unclear
whether the text describes an eclipse at all. In this class, the astronomical
dating of written evidence is either senseless or allows for so many possible
astronomical solutions that they all fall into practically any of the prescribed
historical epochs.
(2) Detailed evidence. In this class, an astronomical solution often turns out
to be unambiguous (or there are two or three solutions).
2.4. The effect of shifting the dates of eclipses forwards
One important fact discovered in [2] is that all the eclipses of the second class
are not dated by the formal astronomical approach traditionally as, for ex-
ample, in [8] and [12], by 700 B.c. to A.D. 400, but substantially later . These
dates sometimes differ from the traditional ones by several centuries. Mean-
while, these new astronomical solutions (dates) fall into the time interval from
A.D. 400 to 1600. The forceful distortion of the dates, made by the earlier chro-
nologists, and fixed in the classical papers [8] and [12], is due to the pressure
of chronological tradition. The astronomers had to look for the required astro-
nomical solutions only within a narrow, prescribed time interval. More than
that, in most cases, an exact astronomical solution could not be found, and
the astronomers had to exhibit an eclipse only partly satisfying the descrip-
tion contained in the ancient document (see the examples below). In making
use of the method of formal astronomical dating, this deviation (strained so-
lution) vanishes, which leads to the appearance of new astronomical solutions
(dates), although different from the traditional ones.
Continuing the investigations started in [2], the author has also analyzed
the medieval eclipses from A.D. 400 to 1600 on the basis of formal astronom-
ical dating methods. It turned out that the effect of shifting the eclipse dates
forwards, which had been discovered in the above-mentioned paper only for
ancient eclipses, could be extended also to the eclipses traditionally dated as
belonging to the interval a.d. 400-900. In particular, many pieces of writ-
ten evidence (due to the extreme vagueness of their formulations) proved to
admit a large spectrum of astronomical solutions distributed over the entire
possible period of history. It is only beginning with about A.D. 900, and not
A.D. 400, as was suggested originally [2], that the traditional eclipse dates be-
come satisfactorily consistent with the results of the application of the formal
astronomical dating method. Finally, this consistency becomes reliable only
after about A.D. 1300.
This result agrees with the theory of empirical corrections employed in ([8],
pp. 4-6), to revise the formulas for the calculation of eclipse dates. Thus, as
21 “basic eclipses” receiving detailed descriptions in at least 10 ancient texts,
we can take those dated to the right of (later than) A.D. 840 (42 reports)
and distributed in the interval up to A.D. 1386 [8]. On the other hand, we
20
Problems of Ancient and Medieval Chronology
Chapter 1
recall that it is since a.d. 1300 ([12], p. 114) that the graph of parameter
D " has finally been stabilized and aligned. Thus, this moment of the graph’s
stabilization coincides with the origin of the interval of reliable consistency
of the dates of ancient eclipses and with the results of the application of
the formal astronomical dating method. An extremely small number of the
dates of eclipses in the time interval from A.D. 400 to 900 we discovered and
which do not contradict the formal astronomical dating method, is of little
importance statistically. At any rate, this is true from the standpoint of the
new computation of the graph of D ,f .
Before stating the results of the new computations of the graph of D " ,
we shall dwell at length on the effect of shifting the dates of ancient eclipses
in the Middle Ages. Above, we have indicated the difficulties facing the as-
tronomers in dating many of the eclipses traditionally. They were caused by
the requirement of traditional chronology to place the dates of eclipses in a
narrow, prescribed time interval. The formal astronomical dating method re-
moves these obstacles. Because of considerable factual material, we give here
only a short summary, the final results, and typical examples.
2.5. An example: three eclipses of Thucydides
Example 1. Consider the three famous eclipses of Thucydides (the so-called
triad; see [8], pp. 176-179, eclipses 6, 8, 9). They are linked into one triad
by their having been described in one historical text, namely the History
of the Peloponnesian War (Bks. II, 27-28; IV, 51-52; VII, 18-19, 50). The
descriptive characteristics of the triad, which are extracted from Thucydides’
text unambiguously, are of the following form.
(1) All three eclipses were observed in the Mediterranean region, namely,
in a square approximately bounded by the longitudes 15° E. and 30° E. and
the latitudes 30° N. and 42° N.
(2) The first eclipse was solar.
(3) The second eclipse was solar.
(4) The third eclipse was lunar.
(5) The time interval between the first and second eclipses was 7 years.
(6) The time interval between the second and third eclipses was 11 years.
(7) The first eclipse occurred in summer.
(8) The first (solar) eclipse was total (since “the stars were visible”), i.e.,
its phase $ is 12".
(9) The first eclipse occurred in the afternoon (local time).
(10) The second (solar) eclipse occurred at the beginning of summer.
(11) The third (lunar) eclipse occurred at the end of summer.
(12) The second eclipse occurred approximately in March.
Condition 12 is not clearly determined from Thucydides’ text and, therefore,
is not included in the final list of conditions.
The problem arises to find a triad of eclipses completely satisfying all condi-
tions 1-11. Work [8] gives the traditional astronomical solution, namely, 431,
424, and 413 B.c. However, as has been known long ago, it does not satisfy
§2
The Moon’s Elongation and Ancient Eclipses
21
all the data of the problem. As a matter of fact, the eclipse of 431 B.c. was
not total as required by condition 8. It was only annular with phase 10" for
the observation zone and could not be observed as total anywhere on the
earth’s surface ([8], pp. 176-177). This important circumstance was noted by
many authors, e.g., J. Zech, E. Heis, N. Struyck, G. Riccioli, F. Ginzel, and
I. Hoffman [8]. A considerable number of astronomical papers were devoted
to the recalculation of the phase $ of the eclipse of 431 B.C., for which various
admissible corrections were introduced into the equations of the lunar theory
in order to make the phase close to 12". Thus, Dionysius Petavius obtained
$ = 10^25 for the observation zone, Struyck 11" ([8], p. 176), Zech 10"38 [14],
Hoffman 10"72 ([8], p. 176), and Heis even 7"9(!) ([8], p. 176). In the modern
literature, the phase value is assumed to be 10" [8]. We stress once again that,
due to its annular form, the first eclipse in 431 B.C. was total nowhere on earth
for any latitude and longitude. Accordingly, Ginzel wrote 2 :
“The insignificance of the eclipse phase was somewhat shocking. . . . According to
the new calculations, the phase was equal to 10". . . . ” [8].
Besides, certain other conditions were not fulfilled either. For example, the
umbra passed through the observation zone only after 17 hours local time,
and even after 18 hours according to Heis [8], which means that condition 9
(the eclipse occurring in the afternoon) is satisfied only approximately.
Certain authors ([8]; see the survey) carried out the calculation of the co-
ordinates of bright planets, thinking that they could have been seen during
the annular eclipse, in order to satisfy the important condition 8. However,
the obtained results showed clearly that the planets’ positions on the celestial
sphere during the eclipse of 431 B.c. did not provide for their reliable visibility.
If Venus could have been visible, then, for example, Mars was only 3° over the
horizon (Heis’s computation), while Jupiter and Saturn were below the hori-
zon, and so forth ([8], p. 177). Johnson suggested another astronomical solution
for Thucydides’ first eclipse, namely, 433 B.C. Although it soon became clear
that this solution still did not satisfy the data of the problem posed, it was now
for other reasons [8]. Besides, this eclipse had a short phase, namely, 7" 8 [8].
The largest variation possible of certain constants in the lunar equations,
with the purpose to increase the phase of the eclipse in 431 B.C., was made
by Stockwell. However, it yielded only 11"06 for the observation zone, which
did not account for the completeness of the eclipse either. The computations
were questioned in the literature, too [8].
In this connection, an attempt to revise Thucydides’ text itself, and, in par-
ticular, condition 8, should be noted also. However, its detailed analysis carried
out at the author’s request by E.V. Alexeeva (Faculty of Philology, Moscow
University) showed that the eclipse characteristics were unambiguously deter-
mined from Thucydides. This circumstance had not been questioned earlier,
though.
2 Translated from the German (tr.)
22
Problems of Ancient and Medieval Chronology
Chapter 1
No other astronomical solutions in 600-200 B.c., which would be more
suitable than the traditional solution of 431, 424, 413 B.C., seem to have been
found. It is because of this fact that this incorrect “solution” has been retained
in spite of the above contradiction repeatedly discussed in the literature.
Meanwhile, the application of the formal astronomical dating method and
the extension of the search interval (for astronomical solutions) to 900 B.C.-
A.D. 1600 yield two and only two exact solutions, the first having been given
in paper [2] (Vol. 4, pp. 509, 493-512), while the second one was given by
the author of the present work during the repeated analysis of all the eclipses
from the indicated interval and the construction of their trajectories on the
diagram.
Thus the first solution yields August 2, 1133, March 20, 1140, and August
28, 1151, whereas the second is August 22, 1039, April 9, 1046, and September
15, 1057. Note that the fact of the availability of exact solutions itself is
nontrivial. In both exact solutions found, even condition 12 is fulfilled, the
one not originally included in the list of basic data. Besides, the first eclipse
is total in both solutions (for the observation zone), which is just what was
required by condition 8.
2.6. An example: the eclipse described by Livy
Example 2. Consider eclipse 25 (see [8], pp. 189-190) described in the History
of Rome by Livy (Bk. XXXVII, 4.4). The characteristics extracted from Livy’s
text are as follows.
(1) The eclipse was solar.
(2) It occurred 5 days earlier than the ides of July, i.e., on July 10.
(3) The approximate coordinates of its observation zone were 30° < lat. N.
< 45° and 10° < long. E. < 25°.
(4) In the observation zone, the moon’s trajectory passed below the centre of
the sun during the eclipse if the moon and sun were projected on the celestial
sphere.
The traditional solution suggested in Ginzel’s canon [8] was March 14, 190
B.c. However, since condition 2 was not fulfilled, the astronomers also offered
other astronomical solutions, e.g., July 17, 188 B.C. But the conditions of
the problem posed were not satisfied in this case either ([8], p. 190). Owing
to the absence of other astronomical solutions for the time interval 300-100
B.c., determined beforehand due to the a priori requirements of tradition,
and which would satisfy conditions 1-4 better, the traditional one of 190 B.c.
was retained in the canon.
Meanwhile, the application of the formal astronomical dating method and
extension of the interval in which an exact solution was being squght to the
periods from 600 B.c. to a. d. 1600 permits us to reach the following conclu-
sion [2].
(1) As it turns out, there is an eclipse fully satisfying all the conditions of
the problem.
§2
The Moon’s Elongation and Ancient Eclipses
23
(2) This exact solution is unique for the interval from 600 B.c. to A.D. 1600.
(3) It is July 10, A.D. 967.
(4) It is stable with respect to a small perturbation of the initial data,
namely a perturbation of the principal condition 2, which means that it re-
mains unique in extending the search interval from July 10 to July 9 and 11,
i.e., by one day. This exact solution was found in [2], assuming, naturally, that
the Julian names of the months correspond to the Julian calendar.
2.7. An example: the eclipse described by Livy and Plutarch
Example 3. Consider the list of descriptive characteristics of eclipse 27 (see
[8], p. 190) also described in Livy’s History of Rome, Bk. LIV, 36.1. See also
Plutarch’s Vitae Aemilius Paulus , 17.
(1) The eclipse was lunar.
(2) It occurred on the night of September 4 to September 5.
(3) The observation zone was bounded by lat. 40° and 50° N, and long. 10°
and 25° E.
(4) It occurred from 2 to 4 A.M. local time.
(5) Its phase was close to 12", and possibly exceeded 12".
Remark. The phase of a solar eclipse is found by the formula $ = 12A, where
A is the ratio of the part of the sun’s diameter, covered by the moon at the
eclipse’s maximum, to the whole diameter. However, in the case of a supertotal
lunar eclipse, a quantity proportional to its duration is added to the phase of
12" (the moon stays in the shadow of the earth for a long time). Hence, the
phase of a lunar eclipse can reach 22"7.
(6) This lunar eclipse occurred after the summer solstice.
The traditional solution given in the canon ([8], p. 190) is June 21, 168 B.c.
This does not satisfy conditions 2 and 6 of our problem. Attempts of many
authors to find a better astronomical solution for the interval from 300 to
100 B.c., determined a priori from the requirements of tradition, did not lead
to positive results. Omitting the details, we should note that the situation is
perfectly similar to the one described in Examples 1 and 2.
Application of the formal astronomical dating method and extension of the
search time interval from 600 B.c. to A.D. 1600 permit us to draw the following
conclusions [2].
(1) There exist exact astronomical solutions fully satisfying all conditions
1 - 6 .
(2) There are only three exact solutions for the time interval from 600 B.c.
to A.D. 1600.
(3) These solutions are (a) the night of September 4 to September 5, A.D.
415, (b) the night of September 4 to September 5, A.D. 955, and (c) the night
of September 4 to September 5, A.D 1020 ([2], Vol. 5, pp. 266-272).
(4) For a small perturbation of the initial data, i.e., while considering lunar
eclipses occurring not only at night but also at sunset, there arises only one
more possible solution, (d) the night of September 4 to Septemer 5, A.D. 434.
24
Problems of Ancient and Medieval Chronology
Chapter 1
Yet another solution is theoretically possible, namely, the lunar eclipse of
September 4 to September 5, 106 B.c.; however, it possesses the phase 5"9,
which is far too small. If we select those with greatest phases from the above
astronomical solutions (see condition 5 of the problem), then two of them are
ideally suitable, namely the eclipse of A.D. 955 with the phase of 16"1 and
that of A.D. 1020 with the phase of 18"7, the latter being still more adequate
than the former.
Upon further perturbation of the initial data of the problem, namely, also
considering the night of September 3 to September 4, four other new astro-
nomical solutions present themselves; however, they all relate to the medieval
period, occurring in A.D. 453, 936, 1457, and 1476. A perturbation of the ini-
tial data in the other direction, namely, considering the night of September 5
to September 6, is impossible, which follows from Livy’s text.
2.8. An example: the evangelical eclipse described in the New
Testament in connection with the Crucifixion
Example Let us consider the characteristics of lunar eclipse 36 (see [8],
pp. 200-201) described in the New Testament, i.e., the so-called evangelical
eclipse (Mt 27:45, Mk 15:33, Lk 23:44-45).
(1) The eclipse was lunar.
(2) It was related to the spring equinox or occurred on the eve of Passover
(Jn 19:14, 19:30-34). For the present, we do not distinguish between Easter
and Passover.
(3) It occurred on Friday during Passover (Jn 19:14, Mt 27:62).
(4) It lasted for about three hours (Mk 15:33-34).
(5) It lasted from 0 A. M. to 3 A. M. according to the modern count of hours.
This condition is sometimes questioned in the chronological literature; how-
ever, there are valid reasons to believe that the eclipse started approximately
at midnight (see [2]).
The traditional solution given in [8] is April 3, A.D. 33 (the date of the
crucifixion). However, just like in the above examples, this solution does not
satisfy the data of our problem. Namely, although conditions 1, 2, and 3
are fulfilled (the eclipse occurs on the eve of Passover), conditions 4 and 5
are not. In particular, the phase of this eclipse (for the observation zone,
i.e., Jerusalem) is so small that it could have been observed only for several
minutes as the umbra was already sliding off the rim of the lunar disc.
In spite of the quite controversial descriptive characteristics of this lunar
eclipse and the conjecture of some medieval authors and annalists, such as
Synkellos, Phlegon, Africanus, and Eusebius, that it had in reality been solar
(Lk 23:45), we can, nevertheless, endeavor to apply the formal astronomical
dating method. We obtain (see [2]) that the above problem does have at least
one solution.
(1) In the time interval from 200 B.c. to A.D. 800, there really exists a lunar
eclipse satisfying conditions 1-5.
§2 The Moon’s Elongation and Ancient Eclipses 25
(2) In the time interval from 200 B.C. to A.D. 800, this astronomical solution
is unique.
(3) The solution is March 21, A.D. 368. The (supertotal) eclipse phase is
large and equals 13"3.
It should be noted that the calculations with the purpose of discovering
a suitable lunar eclipse were made in the interval only up to A.D. 800 [2].
As a matter of fact, Morozov believed that the related historical events (the
crucifixion) could not have occurred later than A.D. 800 ([2], Vol. 1, p. 97).
Without imposing this restriction a priori , we extended the calculations along
the time axis to embrace the whole interval up to A.D. 1600.
The author has therefore found only one more possible astronomical solution
of the problem: the eclipse of April 3, A.D. 1075. There are no other solutions.
The eclipse of A.D. 1075 did occur on Friday on the eve of Passover, which
was on April 5, 1075. However, its phase was small, $ = 4"8. The moment
when half of the period of the eclipse elapsed was at 23 hours and 18 minutes
GMT, which means, in particular, that, in the latitude of Rome, for example,
it occurred at about midnight. The coordinates of the zenith were lat. 10° N
and long. 8° W. Its date (April 3) coincides with the canonical one of the
evangelical eclipse (see, e.g., [8], p. 200). The date April 3 is regarded as
canonical and traditional. Besides, the lunar eclipse of A.D. 1075 occurred
precisely on the eve of Passover, which is consistent with the requirements of
the tradition that assumes that the crucifixion occurred on the eve of Passover.
Recall that the traditional astronomical solution of April 3, A.D. 33, is also
two days prior to Passover, which occurred on April 5, A.D. 33.
2.9. The oscillation of a new graph of D" about one and the same
value. No nongravitational theories are necessary
We now list certain results. If *£ci w denotes the date of an eclipse obtained
by the formal astronomical dating method, and is the traditional date
given, e.g., in the canons of Ginzel and Oppolzer [8], [12], then we obtain
the following result. As it turns out, for all the eclipses of the second class
(i.e., those thoroughly described in ancient texts), the following important
inequality is valid, namely,
.iold jnew
l ecl. > l ecl. *
Moreover, this shifting of the dates of ancient eclipses of the second class
forwards in time (see above) is carried out in the following uniform manner.
All eclipses of the second class traditionally dated in the interval from 900 B.c.
to A.D. 400, turn out to be mechanically shifted further ahead than A.D. 400
into the Middle Ages. It happens that the percentage of eclipses of the first
class (inaccurately and vaguely described in the sources) is extremely high
from a.d. 400 to 900. The dates of such eclipses are either almost incalculable
by astronomical means due to the imprecision of written evidence about them
or are shifted forwards again. Starting with A.D. 900, the new dates are
satisfactorily consistent with t°l f . Only beginning with A.D. 1300 does this
26 Problems of Ancient and Medieval Chronology Chapter 1
correlation get quite reliable. The complete picture of shifting the dates of
ancient eclipses forwards on the time axis turns out to be rather confusing
and complicated because of the nonuniqueness of astronomical solutions. Nev-
ertheless, it obeys a certain law which we shall describe below.
Recall that the behaviour of the graph of D"(f), computed by R. Newton
on the basis of the prior eclipse dates , also makes it possible to naturally
distinguish certain time intervals characterized by an essentially different be-
haviour of the parameter D N on the time axis. Remember that the values of
D u oscillate about zero (though very few of them have been computed) ear-
lier than a.d. 400. Then, in the time interval A.D. 400-1000, a considerable
chaotic variance of the parameter values is noticeable. Finally, since A.D. 1000
(and still more since A.D. 1300), the values of D" are already close to those
known today. Thus, we obtain one important conclusion. It turns out that
the time limits discovered for the behaviour of D"{t) almost coincide with the
characterization of different shifts of the dates of ancient eclipses forwards on
the time axis. This indicates a possible relation between the two important
effects, namely, (1) the square wave in the behaviour of D H and (2) the shift
of the dates of ancient eclipses forwards due to the application of the formal
astronomical dating method.
Let us ascribe to each ancient eclipse from historical sources its new date
*eci w calculated by the formal astronomical dating method. The recalculation
of the values of parameter D " on the basis of these new dates, which I carried
out in [4], is shown in Fig. 7.
Figure 7. New curve for D n (t) in the interval from 900 B.C. to A.D. 1900
(A.T. Fomenko).
We see that the replacement of the date by t™i not only shifts the his-
torical events and texts describing the eclipses forwards in time, but also,
in most cases, leads to the identification of “former eclipses” (previously
§2
The Moon’s Elongation and Ancient Eclipses
27
regarded as ancient) with medieval ones known from other sources. More-
over, the “ancient eclipses” overlap with the medieval ones, many of which
were used by R. Newton for the computation of the former graph of D" for
the period from A.D. 400 to 1900, thus adding new texts previously treated
as antique and traditionally dated as older to the formerly medieval one, with
information about medieval eclipses being used to determine the “medieval
part” of the graph of D " . Therefore, we now have to take into account these
new additional data (characteristics of eclipses) which earlier were ascribed
to other, presumably ancient eclipses in recalculating the medieval part of
the graph of D n . A priori, such an extension of the list of descriptive char-
acteristics of certain medieval eclipses could have led to a contradiction with
their characteristics known earlier from medieval texts, and, in particular, to
a change of the formerly medieval values of D H (t) from A.D. 400 to 1900.
However, the detailed investigation of all descriptive characteristics, both old
and new, has shown that the formerly medieval values of D " from A.D. 400
to 1900 are almost unaltered.
Based on this result, we can make the following conclusions: The new curve
of D n from A.D. 400 to 1900 practically coincides with the former. From 900
B.c. to A.D. 400, the new curve of D" is simply undetermined, since no reliable
eclipse dates exist for this time period.
The new graph of D n is qualitatively different from the former. We see that
D N varies along a smooth and nearly constant curve (horizontal in Fig. 7)
which oscillates about one and the same value — 18"/century 2 from A.D. 900
to 1900. The parameter D" undergoes no sharp change. It invariably retains
approximately the modern value. No nongravitational theories of the type sug-
gested in [9] are therefore necessary. It is interesting that the variance of the
values of D n , quite insignificant from A.D. 900 to 1900, gradually increases in
shifting to the left from A.D. 900 to A.D. 400. In our opinion, this fact indicates
the vagueness and insufficiency of the observational data contained in ancient
historical texts describing this period. Then (see Fig. 7), to the left of A.D. 400,
the zone starts where reliable observational data (which may have survived to
the present day) are absent, which reflects the natural distribution in time of
astronomical observational data supplied by the ancient chroniclers. Appar-
ently, the exactness of observational data and textual descriptions from A.D.
400 to 1000 was extremely low. The precision of observations and descriptions
started to improve afterwards as the technology and the instruments improved
and became more sophisticated, which is reflected by the gradual decrease in
the variance of the values of D N . Finally, in the era of developed astronomy,
we see that the curve of D n is all but aligned, and stable from A.D. 900 to
1900.
All the previous results show that the dates of ancient eclipses are shifted
forwards in their redating, with the magnitude of the shifts being expressed
by the positive quantities t — tf ^ .
28 Problems of Ancient and Medieval Chronology Chapter 1
2.10. Three rigid “astronomical shifts" of ancient eclipses
We see that, for different eclipse groups previously regarded as ancient, the
values of the shifts forwards are much different, which leads to a great con-
fusion in the general picture of redating ancient eclipses. Nevertheless, it so
happens that the system of redating and shifting the dates forwards in time,
which appears rather chaotic at first glance, allows the derivation of an im-
portant regularity, discovered by the author. This law completely agrees with
the system of the three rigid chronological shifts in ancient history, discovered
by the author in [5] on the basis of quite different statistical investigations
(of nonastronomical character), which we shall not define here due to lack of
space. However, we formulate only the final result, while referring the reader
to the aforementioned work.
Consider the historical time interval from 1600 B.C. to A.D. 1700. Cover
it with a system of line-segments denoted arbitrarily by the letters K t H ,
n, C, T. Placed on the time axis, they are of different length, and the same
letter, or interval, can be repeated several times. Moreover, different intervals
designated differently can overlap one another, in which case we will denote
the letters of the overlapping intervals by “fractions”. In Fig. 8, they are
represented for better visualization by different geometrical symbols such as
rectangles, triangles, and trapezia. The covering of the obtained time axis will
be referred to as the global chronological diagram (GCD). It has the following
form:
GCD = T KTHTTKTHTKTTKTHTTPTC
P C P n S ~C
~~C P
P
We will also indicate the end-(boundary-)points of all these intervals, for
which we rewrite the chronicle line in the GCD, marking the dates of the
beginning and end of each epoch interval. For example, the symbol 1570 T
1550 means that the given copy of the epoch interval T starts in 1570 B.C.
and ends in 1550 B.C. If the left-hand number is greater than the right-hand
one, then we mean the years B.C.; otherwise, the years A.D. are meant. Thus,
we have:
GCD = 1570 T 1550, 1460 K 1240 T 1226 H 850 T 830, 760 T 753 — 523,
523 T 509 - 82 T 27 27 B C - K 217 A ^ 2 17 T 251,
500 C 250 82 B.C. P 217 A.D.
306 K 526 roc -TO rro 552 H 915
333 H 526 526 1 552 661 II 867
270 T 306
962 P 1254 T 1273
240 § 580
962
950 C 1300
650 P 900
1273 C 1619.
915 T 925, 932 T 954,
§2 The Moon's Elongation and Ancient Eclipses 29
Figure 8 . Statistical duplicates in European history and biblical history (shifted
forwards ). Three chronological shifts.
30
Problems of Ancient and Medieval Chronology
Chapter 1
1
8
1
8
CM
I
8
CO
3
&
op
CM
in
*
8
+
8
-I-
1
in
£
+
1
§
+1619
■
T
K
T
H
T
T
K
P
T
X 1 O
T
K
P
T
T
K
n
c
p
T
T
T
p
C
T
C
■
D
D
D
D
D
D
D
B
C«
1.778-year shift
K
T
H
T
P
T
c
C,
1.053-year shift
K
T
D
B
B
B
B
B
fl
fl
Q ,
333-year shift
1
1
i
B
1
B
fl
C
c,
Different letters represents different periods
B
B
B
B
■
B
B
B
c*
K
1
1
1
1
B
B
C
Co
Figure 9(1). Formal decomposition of the “ modern textbook” of ancient his-
tory into the sum of four “isomorphic fibres”.
The identical letters indicate the epoch duplicates. For example, the epoch
500 C 250 duplicates 950 C 1300. The GCD resolves into the composition
of the three rigid shifts shown in Fig. 9(1). By adding the horizontal lines
vertically (i.e., by superimposing them) and by identifying the same letters
placed over each other in the same column, we obviously obtain the complete
GCD. Thus, we can write symbolically that the GCD = C\ + C 2 + C3 4 * C4 ,
where the chronicle line C\ equals Co 4 - C". All four chronicle lines Ci, C 2 , C 3 ,
and C 4 are almost indistinguishable, i.e., they consist of the same sequences
of letters. The chronicle C 2 is meanwhile glued to C\ with a backward shift of
about 333 years, C3 to C\ + C 2 with a backward shift of approximately 1,053
years, and C4 to C\ + C 2 + C 3 with a backward shift about 1,778 years in
length. All three basic rigid shifts are counted from one and the same point
(Fig- 7).
We obtain that the three chronological shifts discovered on the basis of
statistical methods of non astronomical character (see the author’s paper [5])
§2
The Moon’s Elongation and Ancient Eclipses
31
manifest themselves also in a rather complicated picture of a forward shift of
the revised ancient eclipse dates. It should be noted, as can be seen in the
above examples, that for many ancient eclipses, there usually exist several
different astronomical solutions, all “enjoying equal rights” . Hence, it is ap-
parent that the inverse problem, namely restoration of the three chronological
shifts on the GCD based only on astronomical data, cannot be solved reliably
today.
2.11. The complete picture of astronomical shifts
We can now outline the complete picture of shifting the redated ancient eclipse
dates forwards.
If the traditional date of an eclipse is represented by a point in the
time interval from 1600 B.C. to A.D. 1700, then it will necessarily fall on
one of the chronicle lines Ci, C 2 , C 3 , and C 4 , which make up the GCD. C 4 ,
C 3 , and C 2 are moved forwards in the inverse shift by 1,778, 1,053, and 333
years, respectively. Meanwhile, they are again identified with the chronicle
C\ embracing the events from a.d. 300 to 1619. The following important
statement turns out to be valid.
The forward shift of an astronomically redated eclipse by the quantity t^ w —
tf$ usually coincides with that of one of the chronicles C 2 , C 3 , and C 4 into
which the date fell originally. However, the value of sometimes
coincides with the difference or sum of certain of the three basic chronological
shifts. In other words, the astronomical and chronological shifts are consistent.
Thus, all the dates of astronomically redated ancient eclipses are broken
into certain groups, each of which is shifted forwards by about 1,778, 1,053,
and 333 years, or their difference or sum. It is important that the relative posi-
tion of the dates t^ inside each of these groups is then retained qualitatively
and that the eclipses fall into the time interval A.D. 400-1800.
This shift of the dates of ancient eclipses can also be described thus: If an
eclipse has fallen into some copy of the interval P on the GCD, then, redated
astronomically, its new date f"|j w is shifted forwards and gets into one of the
other replicas of P to the right of the traditional date . Further, it turns
out that the majority of the dates of eclipses previously regarded as ancient
are shifted into the time interval A.D. 800-1700 [5].
2.12. The coincidence of the astronomical shifts with the three basic
chronological shifts in the global chronological diagram
We illustrate our statement regarding the coincidence of the astronomical
shifts, arising in redating the eclipses, with the three basic chronological shifts
on the GCD by several typical examples.
(1) The triad of eclipses described in Thucydides* History of the Pelopon-
nesian War (see Example 1 above) is closely related to the historical events
in ancient Greece, traditionally dated for about the 5th century B.C. [ 8 ], [12].
As we have noted, the first eclipse mentioned by Thucydides is placed by
32
Problems of Ancient and Medieval Chronology
Chapter 1
traditional chronology according to the ancient sources at around 431 B.c.
[8]. Redated astronomically, its date was shifted forwards, and the eclipse was
identified (see the second astronomical solution in Example 1) with the me-
dieval one of A.D. 1039. Therefore, the value of the astronomical forward shifts
of the eclipse date is approximately equal to 1039 + 430 = 1469 years, which
is close to the difference of two basic chronological shifts on the GCD, namely,
1778 - 333 = 1445 years.
(2) The lunar eclipse of Livy from Example 3 is closely related to the events
in ancient Rome, which occurred, according to traditional chronology, in the
middle of the 2nd century B.c. [8], [12]. Astronomically redated, it was also
shifted forwards, and the eclipse was identified with the medieval one of A.D.
955, the first astronomical solution in Example 3. Thus, the astronomical shift
forwards is approximately 1,100 years, which is close to the basic chronolog-
ical shift on the GCD of 1,053 years. The shift makes the chronicle line C$
coincident with C\.
(3) The lunar eclipse of Example 4 (see above) is inseparably linked with
the events (the crucifixion) occurring, according to the traditional chronology,
in about A.D. 33 [8].
Astronomically redated, it was shifted forwards, too, and identified either
with that of a.d. 368 or 1075. In the former case, the astronomical shift is 335
years, which all but exactly coincides with the basic chronological 333-year
shift of the chronicle line C 2 on the GCD; in the latter, the shift forwards is
about 1075 — 33 = 1042 years long, which is close to the astronomical shift by
1,053 years.
We should not think that the three astronomical shifts discovered reflect
any periodicity in the distribution of authentic eclipse dates belonging to
the past. As a matter of fact, in case (1) in Example 1, the astronomically
exactly calculated date is only A.D. 1039. The first and traditional date
of 431 B.c. is astronomically inexact and appears only under the pressure
of chronological tradition, which a •priori relates the historical events linked
with Thucydides’ eclipses to the 5th century B.C. However, the above date of
431 B.C. was obtained by the astronomers as the result of a forcibly strained
calculation in order to satisfy the requirements of tradition [8].
§3. Traditional Chronology of the Flares of Stars and the Dating of
Ancient Horoscopes
3.1. Ancient and medieval flares of stars. The star of Bethlehem
The important fact is that the three discovered chronological shifts on the
GCD are very consistent with many other astronomical data not yet linked
to the eclipses directly. For example, I have analyzed the traditional chro-
nology of the flares of the so-called novae and supernovae. Let us list the
flares of all such stars regarded as reliable in accordance with [3], [6]: 2296
and 2241 B.c., a.d. 185, 393, 668, 902, 1006, 1054, 1184, and 1230, and those of
the 16th century (see Kepler’s list). The so-called star of Bethlehem described
§3 Flares of Stars and the Dating of Ancient Horoscopes 33
in the New Testament (Mt 2:3), i.e., a flare occurring in about A.D. 1, is also
usually added. The study of the astronomical situation in about A.D. 1 with
the purpose of discovering the remains of this famous “star” was taken up, for
example, by J. Kepler and L. Ideler (see [1], pp. 128-129). In the chronological
shift backwards by 1,053 years (which corresponds to the superposition of the
chronicle C 3 on C\ + C 2 ), the time interval A.D. 962-1250 is placed on the
time interval 91 B.C.-A.D. 197 (see the chart).
Ancient History from 91 B.c. to a.d. 197
Medieval History in A.D. 962-1250
The complete list of the flares of stars
in this epoch, fixed in antique sources,
is:
the famous flare in a.d. 1;
the flare in A.D. 185.
The complete list of the flares
of stars in this epoch, fixed in
medieval sources, is:
the flare in A.D. 1006;
the famous flare in A.D. 1054;
the flare in A.D. 1230.
(1) The famous “star” of A.D. 1 when
Christ was born (Mt 2:2, 8, 9-11).
(1) The famous flare in a.d. 1054
of the supernova in Taurus.
Hildebrand was “born” as
reformer of the Church.
The dates of these flares are ideally coincident in shifting by 1,053 years
(see the GCD).
(2) The flare in A.D. 1 was visible
“in the east” (Mt 2:2,9) (“The star
which they saw in the east”).
(3) The flare of the star in A.D. 185.
(2) The flare of the star in A.D. 1054
was visible “in the eastern sky” [ 6 ].
(3) The flare of the star in A.D. 1230
The dates of the flares in A.D. 185 and 1230 can also be made coincident
under the same 1,053-year chronological shift with a difference of only 8 years.
(4) The star flare in A.D. 185 (4) The star flare in A.D. 1230
lasted 7 months [3], [ 6 ]. lasted 6 months [3], [ 6 ].
Thus, the dates (regarded as trustworthy) of all star flares from 900 B.c.
to a.d. 390 are obtained from those of the medieval flares of stars from the
10th to the 13th century under the backward 1,053-year chronological shift.
This new independent corroboration of the existence of the global 1,053-year
chronological shift is interesting, since we have analyzed here the dates given
by the written sources of quite irregular astronomical phenomena. Note that
the previous traditional dating of these flares was carried out on the basis of
a written chronological tradition of “nonastronomical origin” .
34 Problems of Ancient and Medieval Chronology Chapter 1
3.2. Astronomical dating of ancient Egyptian horoscopes
We now turn to an analysis of the astronomical results of [2] of dating the
zodiacal positions of the planets, which are described in certain historical
sources, namely, the so-called horoscopes. Recall that all planets are placed
near the ecliptic, relative to the stars (i.e., on the fixed astral sphere), and
their position can be calculated similarly to the method of determining the
dates of ancient eclipses. That is, we have to fix the positions of observable
planets relative to the zodiacal constellations at some modern moment of
time. Then, plotting integral multiples of the (known) sidereal periods of the
planets backwards, we can, in principle, calculate horoscopes of the past, i.e.,
the position of the planets relative to the zodiac at a prescribed moment of
time.
Thus, if a horoscope is described in some historical source, then, proceeding
analogously with the procedure of calculating ancient eclipse dates, we may
attempt to date it. To this end, we have to compare its description in a
historical text with the calculated tabular horoscopes and attempt to find a
horoscope with the same characteristics.
The seeming simplicity of this idea is made very complicated by the diffi-
culties of the calculations and, which is most important, by various secondary
reasons of “nonastronomical” character similar to those with which we are
already familiar.
In [2], Morozov analyzed the traditional dates of all the basic horoscopes
fixed in the surviving ancient sources. Omitting the details, we inform the
reader that the result was the same forward shifts of their dates obtained
astronomically as occurred previously in the case of ancient eclipses. We give
a typical example.
The well-known Egyptologist W. Flinders Petrie in 1901 discovered in Up-
per Egypt (Athribis) an ancient Egyptian interment dated by the traditional
chronology from the 1st century B.c. to the 1st century A.D. The interment
was found to contain two graphic images of the planets on the zodiac. The two
horoscopes probably indicated the dates of the two tombs. The specialist Kno-
bel [7] attempted to date the horoscopes within the a priori time interval from
the 1st century B.c. to the 1st century A.D. However, no exact astronomical
solution was found. We make the precise statement that the a priori interval
was determined, proceeding from the style and character of the inscriptions
in the grave, due to which Knobel was forced to offer only quite approximate
values, namely, A.D. 52 and 59. Knobel noted the imprecision, because the
position of Venus at that time was different from its representation in the
tombs.
Then the Russian astronomer M.A. Vilyev analyzed all the horoscopes from
500 B.c. to A.D. 600; however, he discovered no exact astronomical solution for
the Athribis horoscopes. Nevertheless, the extension of the search time interval
and the application of the formal astronomical dating method led Morozov
to the discovery of an exact astronomical solution, namely, A.D. 1049 and
§3
Flares of Stars and the Dating of Ancient Horoscopes
35
A.D. 1065 ([2], Vol. 6, p. 745). It is important that it is unique in the whole
historical interval.
3.3. Astronomical dating of the horoscope described in the Book of
Revelation
Consider another example. An exact astronomical solution for the horoscope
described in the Book of Revelation was suggested by Morozov [2]. Though its
descriptive characteristics can be extracted from the Book of Revelation only
with some controversial interpretation, this circumstance does not encumber
the application of the formal astronomical dating method. As it turns out,
there are only two exact astronomical solutions in the whole historical time
interval, namely, A.D. 395 and A.D. 1249, although the latter was rejected as
“too late” ([2], Vol. 1, p. 53). Besides, it is less satisfactory astronomically.
My analysis of the whole collection of horoscope datings given in [2] has
shown that their forward chronological shift obtained by the formal astro-
nomical method is also due to the same three basic ones on the GCD (Fig. 8).
For example, the forward shift of the dates of the two Athribis horoscopes
is approximately 1,000-1,050 years. Recall that traditional chronology dates
them to about the beginning of the present millennium. Thus, the astronom-
ical shift is close to the chronological one of 1,053 years (see Fig. 67 in Vol. 2
of this book).
The same situation occurs in our last example, that of the Book of Reve-
lation. The second of the two astronomical solutions for its horoscope, A.D.
1249, yields the forward shift of its creation by about 1,050-1,100 years. Note
that the approximate traditional date of writing the Book of Revelation is,
according to A. Harnack, Eberhard, J. Martineau, and van Eising, the second
half of the 2nd century A.D. In this case, the value of the astronomical shift
is thus also close to that of the chronological one by 1,053 years (Fig. 8).
In conclusion, we indicate another interesting astronomical fact [2], which
also turned out to agree with the discovered decomposition of the GCD into
the sum of three shifted chronicles.
The first Latin edition of Ptolemy’s famous Almagest (published in a.d.
1537 in Cologne) contains a catalogue of stars with the indication of their
longitudes and latitudes, i.e., coordinates on the celestial sphere. As is clear
from the text, the catalogue was made by Claudius Ptolemy himself in the
second year of the rule of the Roman emperor Antoninus Pius, traditionally
related to A.D. 138-161. It turns out that there exists a reliable method to
determine from its star catalogue the date when the Almagest was written [2].
Since it contains ecliptic star coordinates, we can make use of the generally
known property of stars to annually increase their longitudes by 50"2 (due
to precession). Dividing the difference between the longitudes indicated in
the Latin edition and those of the present day by 50"2, we shall obtain the
required date. This simple calculation unexpectedly shows that the longitudes
of the stars listed in the first Latin edition of the Almagest were observed or
36
Problems of Ancient and Medieval Chronology
Chapter 1
Figure 9(2). Portrait of the a Imperator Caesar Diuus Maximilianus Pius Felix
Augustus ” (A. Direr).
recalculated by its author in the 16th century A.D. Hence, these astronomical
data belong to the time when the book was published.
Morozov’s work ([2], Vol. 4), supplies many other arguments in favour of
the conjecture that this text was created by the astronomers of the 10th to the
16th century A.D. In our case, the forward shift of the date of writing the Al-
magest is about 700 or 1,390 years if obtained astronomically. Meanwhile, we
compare the date A.D. 1530 (epoch of the first editions of the Almagest) with
A.D. 140 (second year of Antoninus Pius’ rule). We obtain 1530 — 140 = 1390
years. The value of this shift is also completely consistent with the GCD
§3
Flares of Stars and the Dating of Ancient Horoscopes
37
(Fig. 7), since it practically coincides with the sum of the two basic chrono-
logical shifts: 1053 + 333 = 1386 years.
In such a forward shift of dates, the period of Antoninus Pius’ rule falls
into the epoch when the first editions of the Almagest appeared, namely, a.d.
1528, 1537, 1515(7), 1538, 1542, and 1551. Note, in conclusion, that immedi-
ately before this medieval epoch, the emperor Maximilian I Pius (!) Augustus
(a.d. 1493-1519) had ruled in the medieval Empire of the Hapsburgs. It is
interesting that he was a contemporary of A. Diirer, the creator of the astro-
graphic charts that accompanied Ptolemy’s Almagest. The prints were made
by Diirer in about a.d. 1515. Therefore it cannot be excluded that it was un-
der Maximilian Pius that the astronomical observations fixed in the Almagest
were carried out (Fig. 9(2)). The statistical analysis of the latitudes in the
star catalogue of the Almagest was made in the recent paper [15]. The result
is as follows: the latitudes in the star catalogue of the Almagest were observed
somewhere in the time-interval A.D. 600-1300. See also [16].
References
[1] Context Nauka, Moscow, 1978 (in Russian).
[2] Morozov, N.A., Christ. Gosizdat, Moscow-Leningrad, 1926-1932 (in Rus-
sian).
[3] Pskovsky, Y.P., Novae and Supemovae. Nauka, Moscow, 1974 (in Rus-
sian).
[4] Fomenko, A.T., “On the computation of the second derivative of the
moon’s elongation”, in Controllable Motion Problems: Hierarchal Sys-
tems . Perm University Press, Perm, 1980, pp. 161-166 (in Russian).
[5] Fomenko, A.T., “Certain statistical regularities of information density
distribution in texts with scale”, in Semiotika i Informatika , Vol. 15,
VINITI, Moscow, 1980, pp. 99-124 (in Russian).
[6] Shklovsky, I.S., Supemovae . Wiley, New York-London, 1968.
[7] British School of Archaeology in Egypt and Egyptian Research Account.
London, 1908.
[8] Ginzel, F., Spezieller Kanon der Sonnen- und Mondfinstemisse fur Lan-
dergebiete der klassischen Altertumswissenschaften und der Zeitraum von
900 vor Chr. bis 600 nach Chr. Berlin, 1899.
[9] Newton, R., “Astronomical evidence concerning non-gravitational forces
in the Earth-Moon system”. Astrophys. Space Sci. 16, 2(1972), pp. 179-
200 .
[10] Newton, R., Ancient Astronomical Observations and the Accelerations of
the Earth and Moon. Johns Hopkins Press, Baltimore, 1970, pp. 32-47.
[11] Newton, R., “Two uses of ancient astronomy”. Phil. Trans. Royal Soc .,
Ser. A, 276(1974), pp. 99-116.
[12] Oppolzer, T., Kanon der Finstemisse ... Mit 160 Tafeln. K.K. Hof- und
Staatsdruckerei, Wien, 1887.
38
Problems of Ancient and Medieval Chronology
Chapter 1
[13] The Place of Astronomy in the Ancient World : Joint Symposium of the
Royal Society and the British Academy , organized by D.G. Kendall et al.
Oxford University Press for the British Academy, Oxford, 1974.
[14] Zech, J., Astronomische Untersuchungen uber die wichtigeren Finster-
nisse r welche von den Schriftstellem des classischen Altertums erwdhnt
werden f etc . S. Hirzel, Leipzig, 1853.
[15] Fomenko, A.T., Kalashnikov, V.V., Nosovsky, G.V. “When was Ptolemy’s
star catalogue in the Almagest compiled in reality?” Statistical Analysis.
Ada Applicandae Mathematicae 17(1989), pp. 203-229.
[16] Kalashnikov, V.V., Nosovsky, G.V., Fomenko, A.T. “The dating of the
Almagest based on the variable star configurations”. Doklady Akad. Nauk
SSSR 307, 4(1989), pp. 829-832 (in Russian).
CHAPTER 2
New Statistical Methods for Dating
§4. Certain Statistical Regularities of Information Density
Distribution in Texts with A Scale 1
4.1. Text with a scale. The general notion
4.1.1. In the present paper, we list the results of a series of statistical in-
vestigations carried out by the author, and which led to the discovery (for
narrative texts) of new statistical invariants such as the laws of information
density conservation.
By a text X with a scale P, we understand a text (e.g., a narrative one)
endowed with a fixed program plan whose items are numbered by a param-
eter t. Meanwhile, we shall require that X should admit a unique partition
associated with this parametrization (program), and such that (1) to each
value of the parameter t E P f a certain part X(£) of the text may necessar-
ily correspond; and, vice versa, (2) each phrase (or word) of the text T may
necessarily belong to one and only one fragment X(t) for a certain t.
We call such a program plan a scale , or parametrization of the text X.
We will say that the text X is parametrized by t y with X = U X(t), and
X(< 1 )nX(« 2 ) = 0if< 1 #< 2 . teP
Such parametrized texts X are well illustrated by those with historical
character, e.g., historical monographs, chronicles, annals, college textbooks,
diaries, and so forth. In these cases, the part of the parameter t is played
by time, i.e., the dates of the events described (according to a system of
chronology or any other method of their dating).
Still, the above concept of a text with a scale is considerably more general
than in the given example. In particular, the parameter t may range over more
complicated domains than the set of natural numbers (dates of events). For
1 First published as an article in Semiotika i Informaiika , Vol. 15, VINITI, Moscow,
1980, pp. 99-124 (in Russian).
39
40
New Statistical Methods for Dating
Chapter 2
example, t can possess a continuous domain in the case where we take as a text
with a scale the description of some physical continuous process parametrized
by continuous time. Then we can consider the description of the instantaneous
state as a “textual fragment” X(t) when the process of interest is going on.
The description may be “composed” of some system characterizing the process
continuously.
Further, for the applications considered below, the following example is
very important. We can assume that a scale P consists of a sequence of dis-
joint intervals (j 4*, JB*) with integral endpoints. In other words, two intervals
(AkiBk) and ( A C} B c ) are disjoint if k ^ c and the union of all of them makes
up the initial segment (A, J3), with the parameter t ranging over their entire
sequence (see the details in Section 4.2.2).
4.1.2. Let X be a certain historical (or, more generally, narrative) text with
a scale P, represented as the union of the fragments X(t). For simplicity, let
the parameter t range over the positive or negative integers from A to B. For
example, we can assume that X describes the events from the year A to the
year B, though the quantities t y A , and B can then be measured not only in
years, but, say, in months, days, or hours.
4.2. Information characteristics (i.e. 9 informative functions) of a
historical text. Volume function, name function, and reference
function
4.2.1. Now, we associate each value of the parameter t with a set /2(0>
. . . , /jv(t) of the formal information characteristics of part X(t) of a historical
text X, which may describe the events of one year t. As we shall now see, the
quantitative information characteristics /,•(<) can be quite multifarious. We
illustrate this with some basic examples.
Example 1. Let fi(t) = vo1 * ^ , where volX(t) is the number of lines,
pages, signs, or words making up a textual fragment X(t). We then, obvi-
ously, get fi(t) > 0, and Y^A<t<B /i(0 = with V = volX denoting the
total volume of the text X without illustrations, diagrams, or bibliography.
The normalizing condition fi(t) = 1 often happens to be convenient in
various instances of such averaging and comparing and is a discrete analogue
of the normalizing condition in the case of a continuous parameter, namely
Sa M x ) dx = L
Example 2. Let f 2 {t) = where s(2) is the number of references to a
year t in the entire text X, and S that of all the dates (years).
Example 3. Let / 3 (t) = where m(t) is the number of the names of
historical characters mentioned in a textual fragment X(<), and M that of all
the names of all the historical characters from the text X . We shall sometimes
also count the number of references to the names of historical figures in the
fragment X(J), i.e., we shall take into account their multiplicity, or frequency,
of use in the text.
§4
Information Density in Texts
41
Example 4. Let f^{t) = , where mj(f) is the number of references
to some concrete name rrij in a textual fragment X(t ), and Afj is the total
number of references to rrij in the entire text X. In general, this name can be
ascribed to different historical characters.
We shall give other important examples of the functions /,(t) below. We
call them informative, or frequency functions, defined by X.
4.2.2. To determine fi(t) means to formally compute an indicated numerical
characteristic such as volume, the number of names, and so on, of all textual
fragments X(t) of some concrete historical texts X, which may be arranged
so that only the partitions of X into the sum of the fragments X(A*.,Bfc)
are specified uniquely, with X(Ak>Bk) denoting that part of X in which the
events of the interval (A*,#*) with integral endpoints are described. Mean-
while, their description inside X(Ak,Bk) may not be specifically related to
the years composing (A*,#*), due to the convenience of considering an en-
larged scale for a parameter which does not range over separate years but
over separate time intervals, in which case we will denote it by T. It should
be noted that, generally, there can exist several different scales for the same
text X. Thus, the choice of a particular scale in X may vary in accordance
with the problem posed. If a parameter T ranges over a sequence of time
intervals (A*,#*), then the normalized volume function f\(T) is determined
as /i(T) = other words, we compute the average value of
the textual volume function in (A*, B*), where vol(A*, Bk) and vol(A, B) are
the values of the volume function in (A*,B*) and (A,B), respectively. The
“averaged” functions for the other examples listed above are defined similarly.
For example, vol(Afc, Bk) is the number of lines, pages, or words making
up the fragment X(Ajb,B*.) that describes the events in the time interval
(AkiBjc)- Further, vol(A,B) is the total volume of the entire text X repre-
sented as the union of its disjoint fragments X(A*, B*).
Thus, each informative function /,(t) is given by a certain graph defined
on an interval (A,B). All the informative functions under investigation are
nonnegative.
4.2.3. Denote by /(t), or /(t,X), some informative function /,*(t) for a text
X, omitting the subscript i for brevity. Let two (or several) historical texts
X and Y be given, which describe the events in an interval (A, B) of the
history of one region (e.g., state, town, etc.). It is clear that the informative
functions /(f,X) and /(t,Y) constructed for X and Y, respectively, will, in
general, be different. This must be so, since their form is influenced by both
the individual propensities of the authors and the general attitude toward the
events characteristic of the time when the text was written. It is clear that
a monograph on art history and one on military history devoted to the same
period will accentuate various things differently, which can lead, for instance,
to a different distribution of the textual volumes.
A natural question thus arises: How essential are these differences in texts
with a statistical approach to the analysis of their informative functions? In
42
New Statistical Methods for Dating
Chapter 2
other words, can we discover certain general, invariant characteristics of the
informative functions f(t,X) and f(t i Y) i which do not depend (or depend
little) on the authors’ tendencies, and which are determined mostly by the
time interval (A,B) and the region T?
4.2.4. Let us investigate the behaviour of such important characteristics of
the graphs of f(t,X) as the distribution of local maxima (“peaks”). Denote
by f’j(A') those years t of a time interval (A, B) in which the graph of f(t , X)
attains local maxima, i.e., exhibits peaks. Let i vary from 1 to q(X ), where
q(X) is the total number of local maxima.
In the case where the scale of the text X is enlarged (see the above example),
the informative function / is represented by a “steplike” graph. This step
function is constant on each separate interval (A*,#*). It is then convenient
to take its midpoint as a local maximum attained in (A*, 23*)-
Consider the volume function vol X(t) as a basic example. Its value at a
point t equals the volume of a textual fragment X(t), measured in pages,
lines, or words. What is the meaning of “the volume function vol X(t) reaches
a local maximum at a certain point r t (X) in an interval (A, 23)”? It means that
the year r,(X) represented as a point in (A,B) is described in the text X in
more detail, on a greater number of pages, and with more particulars than the
nearby years. What can explain such a difference in the description of different
years? One of the possible (and, probably, basic) reasons for this phenomenon
can be formulated as follows. The author of X possessed more information for
the year r,(J\T) (e.g., historical texts or more extensive documentation) than
for the neighbouring years. It could be for this reason that the events of the
year ri(X) of the epoch (A,B) have been described in more detail.
4.3. A theoretical model describing the distribution of local maxima
for the volume function of a historical text. Primary stock. The
information density conservation law
In Section 4.2.4, we formulated a theoretical model (statistical hypothesis)
describing the distribution of local maxima for the textual volume function.
We now designate by C(t ) the total volume of all the texts written by contem-
poraries about the events of a year t. We shall measure the volume in lines
(pages, words, etc.). It is clear that, in general, we cannot reconstruct the
original form of the graph of C(t ) today, which is due to a loss of the old texts
with time. It is only the part K(t) of the primary information stock C(t) that
has survived. K(t) will be called the surviving information (textual) stock re-
garding the year t. Denote by c,- the years in which the volume graph for the
primary stock C(t) has reached local maxima. It follows that especially many
texts were written in these years (for particular reasons which we do not dis-
cuss here), i.e., the contemporaries have recorded a particularly large amount
of written evidence about them. Let us ask ourselves the following question:
What might be the statistical mechanism of loss and falling into oblivion of
§4
Information Density in Texts
43
textual information which leads to a gradual decrease of the amplitude of the
volume graph for the primary stock C(t )?
We formulate the following hypothesis. Though , with time, the amplitude
of the volume graph for the surviving textual stock decreases gradually (since
ancient texts get lost and destroyed ), more remains from those years c,* whose
events were described by contemporaries in a considerable number of texts.
In other words, the years c< in which the volume graph for the primary
stock C(t) reached local maxima must be close to those points Jb* in which
that of the surviving stock K(t) attains local maxima. In particular , c; and
k{ must be close to those years rj(x) in which the volume graph vol X(t) of
the text X describing the events in a time interval (A, B) has reached local
maxima.
Since the volume graph for the primary stock C(t) is unknown, it is hard
to verify the hypothesis stated in its present form. However, we can verify one
of its important corollaries.
4.4. The correlation of local maxima for the volume graphs of
dependent historical chronicles. The surviving-stock graph
4.4.1. Namely, the years of local maxima r,*(X) should be close to those
denoted by r,(Y) for any two historical texts X and Y describing one historical
epoch (A y B) in the history of the same region I\
In other words, the volume graphs volX(t) and volY(<) must attain local
maxima, i.e., form peaks, approximately at the same points (years) in the time
interval ( A , B).
We call this corollary the F-model. To substantiate it intuitively and in-
formally, we assume that two texts X and Y describe the events in a time
interval (A,B) in the history of one region. Let this period be considerably
removed in time from the authors of X and Y . This means that they are no
longer contemporaries of the described ancient events and, therefore, have to
employ a collection of historical sources surviving from the historical epoch
( A , J3), i.e., the surviving information stock. However, we can assume (without
making a gross mistake) that this surviving textual set (information stock) is
approximately the same for both X and Y. Hence, the volume graphs vol AT(/)
and vol Y (t) must more or less simultaneously reflect the maxima of the graph
of the surviving stock while simultaneously reaching local maxima in ( A,B ).
Thus, a chronicler describes in more detail those years from which more texts
have been preserved.
Recall that the years of the local maximum of the volume graph are char-
acterized by especially much surviving information in comparison with the
neighbouring ones. Another natural corollary to the basic hypothesis will be
given in Section 4.7.2 (the K- model).
4.4.2. We should not think that the amplitudes themselves of the local max-
ima of the volume function (the absolute values of the textual volumes) of two
historical texts describing the same events are close (even with normalizing).
44
New Statistical Methods for Dating
Chapter 2
Simple examples of concrete chronicles show that the values of vol A^r,) and
vol Y (r t ) may be considerably different. Thus, the absolute values of vol X(r t )
at the local maximum points r t * vary upon altering the text X (i.e., depend
on it), which shows that the discovery of the rough “invariants of histori-
cal epochs” should not be based on the absolute amplitudes of informative
functions. The analysis of “fine amplitude invariants” was carried out by S.T.
Rachev and the author.
4.4.3. Let us verify the theoretical F-model against the material of concrete
texts (chronicles, etc.), for which we should first formulate the concept of prox-
imity of two sequences of numbers r,(X) and r,(Y) of local maximum points
for two texts X and Y , namely, their volume graphs can, in general, have
different numbers of local maxima, and q(X) ^ q(Y) (see above). However,
as will now be shown, without loss of generality, we can put q(X) = g(Y),
for which it suffices to assume that certain r,(Y) coincide with some new,
additional local maxima to be added in the case where g(Y) < q(X). In other
words, some maxima of the graph of vol Y(t) are declared to be multiple, i.e.,
we assume that several maxima coincide.
Equalizing the number of local maxima for two volume graphs can be car-
ried out differently and in accordance with the supposedly multiple maximum
points and their multiplicity. We now choose a particular method for equaliz-
ing the number of maxima. In the following, we shall perform the minimization
of the functions in question with the help of all such equalizing methods.
4.5. Mathematical formalization. The numerical coefficient d(X, Y),
which measures the “distance” between two historical texts X
and Y
4.5.1. We now describe the mathematical formalization. The points ri(X)
break the time interval (A, B) into smaller intervals whose lengths are given by
the integers ai,a 2 , . . . , a p , where a\ = r\ — A, a, = r,— r*_i for 2 < i < q(X),
and a p = B — r p _x, their number being p = q(X) + 1. Therefore, we can
define a certain integral vector a(X) = (ai(X),... ,a p (A)) belonging to the
Euclidean space R p of dimension p. Since the sum of all a,- is, obviously, equal
to B — A y the length of (A,B), we can assume that the endpoint of a(A)
lies on a (p - l)-dimensional simplex cr , which can be given by the equation
Ya - i x i = B — A, where a? t * are nonnegative Euclidean coordinates for the
space R p . The simplex a is a closed subset, i.e., with the boundary belonging
to it, which follows from the fact that some a, can be zero (the case of multiple
maxima).
Now, let us consider two historical texts X and Y. We can construct two
integral vectors a{X) and a(Y) with their endpoints on one and the same
simplex <j (Fig. 10). Meanwhile, we assume that both texts describe events on
the same time interval (A, B). Note that we have made use here of the above
remark according to which we can assume that the volume graphs for both
texts have the same number of local maxima.
§4
Information Density in Texts
45
B-A
Figure 10. Method for comparison of two historical texts X and Y. We can
construct and compare two integral vectors.
Consider the difference A of the two vectors a(Y) and a(X). The vector A
belongs to our simplex (Fig. 10). Let A be the usua l Euclidean length of the
vector A. Recall that A = ^/^^ =1 (a t (Y) — a,(X)) 2 .
We now introduce the numerical coefficient d(X,Y) = , where D =
D(X y X) denotes a (p — l)-dimensional ball placed in the hyperplane specified
by the equation x * = B — A. This ball is centered at a(X) and has
the radius A, while D fl <r denotes its intersection with the simplex. If A is
large, then the ball D may contain points located outside the simplex. If,
however, the number A is small, and the point a(X) (the center of the ball)
does not lie on the boundary of the simplex, then the ball D(X, A) is small
and wholly contained in the simplex, in which case we have the equality
Dn<r = D. Finally, vo!5, where S is an arbitrary (p— l)-dimensional subset in
the hyper plane 5Z<=i x » = B — A, denotes either the usual Euclidean (p — 1)-
dimensional volume of the subset S (in which case we will say that we are
dealing with a continuous model) or the number of integral points in the set
S. Recall that a point is said to be integral if its Euclidean coordinates are
integers. In the latter case, we will say that we are dealing with a discrete
model.
46
New Statistical Methods for Dating
Chapter 2
4.5.2. Assume that a text X is fixed. While varying the text Y, it becomes
obvious that the number d(X 1 Y) which we introduced in Section 4.5.1 can
be interpreted (under certain natural assumptions) as the probability of the
point a(Y) getting randomly into the ball D with fixed radius A and with
its center at a(X ). This probabilistic interpretation is not at all essential for
the following and is given here only as a formalism which is useful for the
calculations.
We now introduce on our simplex a (p — l)-dimensional measure in the
following two ways:
Version 1. *S) = ‘he
discrete model (see above).
Version 2. /i(S) = «»***« the continu '
ous model, where 5 is a (p — 1)- dimensional domain (subset) in the simplex cr .
Further, to interpret the coefficient d(X,Y) probabilistically, we need a hy-
pothesis regarding the distribution of the random vector a(Y) on the simplex.
In the simplest case, we assume that a(Y) is uniformly distributed on the
simplex, which means that the probability that it gets into some domain S
equals the measure /i(5). Since /i(<r) = 1, 0 < /i(5) < 1.
Applied to our problem, this conjecture can be confirmed somewhat by the
distribution of the random vector a(Y) being consistent with that of the local
maxima of the volume graphs volY(t) for a variable text Y and a variable
historical epoch (A, B ) with invariable A — B. Since we intend to model the
mechanism of information loss and the fall into oblivion, we should take into
account the circumstance that if an archive of historical documents is lost
during some sort of disaster, or similar circumstances, then it is appropriate
to accept the hypothesis that the destruction of any text from this archive
is “equally likely” . The assumption can be restated as a conjecture regarding
the uniform distribution of the random vector a(Y), where the text Y ranges
over all the texts surviving from all possible historical epochs.
To describe this distribution of the local maximum random vector formally
is unreal for the moment, because we should calculate all the vectors a(Y)
for all surviving historical texts Y, which is, obviously, impossible. We could
certainly investigate a sufficiently large sample from the set of all existing
texts; however, certain complicated problems regarding the representability of
such samples and their homogeneity arise here. Besides, such an experimental
approach could considerably distort the general form of the distribution, since
it is difficult to estimate a priori whether a particular sample from the texts
reflects the real mechanism of information loss. At the same time, the assumed
uniformity of the distribution of the random vector a(Y) well reflects the
circumstance that by altering the historical epoch (A, 5), we “uniformly mix”
all possible reasons for the loss of historical texts.
In certain examples related to other methods of dating (see below), where
the total information requiring quantitative processing is sufficiently clear,
we have experimentally calculated the desired distribution of the random
§4
Information Density in Texts
47
vector. For example, we have found the empirical frequency histogram for the
distribution of the duration of kings 9 reigns.
We would like to emphasize once again that the probabilistic interpretation
of the coefficient d(X,Y ) will not be employed below. First, we carry out an
experimental calculation of the concrete limits within which d(X, Y) varies
for “dependent historical texts” describing the same events. It is the values
of d(X,Y) found experimentally that in the following will serve as a standard
for comparing other pairs of texts under investigation.
The coefficient d(X , Y ) ranges from zero to unity. With the assumed uni-
form distribution, we can say that the smaller its magnitude, the lower the
probability of the random event that the random vector a(Z) distributed on
the simplex uniformly happened to be from the vector a(X) at a distance not
exceeding the observable one between the points (vectors) a(X) and a(y).
If A, the distance between a(X) and a(y), is sufficiently large, then d(X,Y)
may be unity, in which case the above ball D wholly absorbs the simplex cr.
Therefore, no statement regarding a possible proximity of the texts X and Y
can be formulated; these texts are far from each other from the standpoint of
their volume functions.
Note that, for any X and Y describing one historical epoch ( A,B ), we
can always find the same scale P, which permits us (in principle, at least) to
compare even texts of various nature, namely, textbooks, annals, and so forth.
Section 4.6 is of formal mathematical character and may be omitted on first
reading.
4.6. Mathematical formulas for computing d(X 1 Y). Mathematical
corrections of the maxima correlation principle
4.6.1. In this section, we give certain mathematical formulas for computing
and estimating d{X ) Y). If a simplex a is specified by the equation Xi =
s, where all a?,* are nonnegative, then its volume in the continuous model, i.e.,
in the usual Euclidean volume, equals Hence,
^A*-*(r-i)!
where s = B — A, A is the length of the vector a(Y) — a(X), and T is the
classical gamma function. For small A, the inequality turns into an equality.
If B — A, and p and A are all sufficiently small, then we have to make use
of the discrete model to calculate d(X>Y) (see above). In order to do this, we
have to find the number of integral points in the ball D and the simplex a for
the given values of B — A y p, and A. Unfortunately, an exact universal formula
cannot be offered in this case; however, we can derive an asymptotic relation,
which is omitted due to lack of space.
We can estimate this mathematically, beginning with which numerical val-
ues of B — A, p, and A it is possible to resort to the continuous model without
48
New Statistical Methods for Dating
Chapter 2
committing a gross mistake. As a matter of fact, concrete calculations are more
simple within the framework of the continuous than the discrete model. Any
concrete numerical estimation is also omitted. Note, in this connection, that
an interesting mathematical problem arises here: How can an exact boundary
be found, separating the area of application of the discrete model from that
of the continuous one (for a prescribed value of admissible error)?
4.6.2. We now indicate the first mathematical correction to the F-model.
Since, when equalizing the numbers of the local maxima of the volume graphs
of two texts being compared, we are forced to assume certain of the coordinates
a # - of the vector a(Y) to be equal to zero in the case where q(Y) < q(X ), this
process is explicitly equivocal. It is clear that we can take any of a% to a p
as the zero coordinate. We should therefore consider all possible methods of
equalizing the numbers of the maxima and then take the least of all the values
obtained as the coefficient d(X, Y). To eliminate the asymmetry between the
texts X and Y due to the definition of d(X i Y) ) we have to “symmetrize” the
value, i.e., consider d ( x - X )+ d ( Y ' X ) .
4.6.3. Now we present the second correction of the F-model. We have carried
out previous constructions for informative functions of the form f{t , X). How-
ever, we should also consider all their possible smoothings in order to establish
the invariant, characteristic properties of the graphs better. For example, as
the simplest method of smoothing a graph, we can consider the “neighbour-
wise averaging”. For a function /(f), its first smoothing is then constructed as
S l f(t) = the second, 5 2 /(f), as the expression S l (S l f(t )),
and so on. It is evident that this procedure smoothes the graph and eliminates
random, small local maxima. Their corresponding coefficient dj (X, Y) is com-
puted after each stage of the smoothing procedure applied to 5 J /(f, X) and
/(f, Y). As the final value of d(X,Y ), we should again take the least of all
numbers obtained in such a manner. Having calculated d(X,Y) for the origi-
nal and nonsmoothed graphs f(t , X) and f(t , Y), we obtain an upper estimate
of the final value of the coefficient. Note that if we are comparing two texts
X and Y which describe different historical epochs of the same length, then
we should make these intervals coincident on the time axis by superimposing.
4.6.4. The third mathematical correction of the F-model is as follows. For
greater statistical invariance of the obtained results, we should compare not
simply a pair of separate texts X and Y, but two sufficiently large groups of
texts X \ , . . . , X n and Y \ , . . . , Y m , where n and m are supposed to be suffi-
ciently large. Denote the first text group by {X}, and second by {Y}. Then we
have to compare the graphs of the functions f(t , {A }) = f(t , X \ , . . . , X n ) and
/(f, {^}) = /(f , Yi, . . . , Y m ), where /(f, {X}) is the averaged function defined
by the equality /(f,{X}) = ~ /(*> Xi). The averaged graphs f(t,{X})
and /(f, {Y}) so formed have been freed of the random local maxima which
might have appeared in one or the other text for some particular reason of
nonuniversal character. The comparison of the averaged graphs yields a more
reliable statistical picture of the evolution in time of written evidence.
49
§4 Information Density in Texts
4.7. Verification of the maxima correlation principle against
concrete historical material
4.7.1. Let us now verify the F-model against some concrete historical ma-
terial. This model will be confirmed if the coefficient d(X, Y) is “small” for
the majority of pairs of real historical texts X and Y describing the “same”
events in the same time interval (A, B) (we call such texts dependent). On
the contrary, for independent texts, i.e., those describing essentially different
events (or essentially different time intervals), this coefficient must be “large”.
For now, the concept of smallness for the coefficient will not formally be made
precise, since we need more experimental data. Meanwhile, the broad statisti-
cal experience of natural science permits us to estimate (with the assumption
of the uniform distribution of the random vector a(Y)) really small probabil-
ities in problems of this kind. Since the explicit construction (from texts) and
description of informative graphs for concrete historical texts of large size are
sufficiently bulky, we are forced to omit the diagrams here and will give only
some typical examples. We first consider the textual volume function volX(f)
(the volume is measured in pages; see also Section 4.2). For example, we have
taken the monograph of V.S. Sergeev, Essays on the History of Ancient Rome
[1], as the text X , and the famous History of Rome by Livy [2] as the text Y.
The time interval (A, B ) described in both books was taken from 758 to 288
B.C., which means that A = —757, B = —287, and the length of the interval
B — A = 470 years. The “— ” sign denotes the years B.c. The volume graphs
of these texts (with respect to years) are shown in Fig. 11. The continuous
line has been constructed for Sergeev’s text, and the dotted one for Livy’s.
As it turns out, A = 21, p = 14, and the coefficient d{X,Y) measuring the
Figure 11. Volume functions of two dependent texts: V.S. Sergeev y s monograph
u Essays on the History of Ancient Rome v and Livy f s a History of Rome ,t .
50 New Statistical Methods for Dating Chapter 2
proximity of the vectors a(X ) and a(Y) in the statistical sense equals 2 10~ 12 .
The distance between them in the Euclidean sense is 21.
In other words, the probability of random proximity of the texts X and Y
(at the distance A = 21) is majorized by 2 * 10“ 12 . We had obtained it even
before minimizing d with respect to smoothing all the graphs (see Section
4.6.3). Therefore, 2 • 10“ 12 is the upper estimate of the final value of the
coefficient. Being very small, it indicates a considerable correlation between
the historical texts by Sergeev and Livy from the point of view of the volume
graphs. It is but natural, because, describing the “same events”, they are
a priori dependent. The most important is not the small absolute value of
d(X i Y) 1 but the large difference between the values of d(X> Y) for dependent
pairs X,Y and independent pairs X, Y.
Similar results are also obtained for the comparison of the different infor-
mative functions /,*(<, X) and /j(f, Y) for dependent X and Y, which describe
the same events in the same time interval (A, B). For example, consider for the
above historical texts X by Sergeev and Y by Livy the informative function
/ 2 (f,X) = where s(t) = number of mentions of a year t in X . Compare
its graph with the volume graph vol Y(t) for Y. We have thus considered the
time interval from 521 to 294 B.C., i.e., A = —520, B = —293, and the length
B — A = 227. It turns out that A = 8, p = 8. The result of the calculations is
d(volY(t),/ 2 (i,X)) = 510- 6 .
In the latter example, we have smoothed the graphs once, i.e., compared
the graphs of 5 1 / 2 (f, X) and S 1 vol Y(t). Such a small value of the coefficient
measuring the probability with which the vectors a(volY(f)) and a(/ 2 (f, X))
may randomly be at the distance A = 8 explicitly expresses the correlation
between different informative functions of dependent texts describing the same
period in the history of the same region or state.
These results indicate the existence of another special law of information
density conservation to be formulated as a K- model below.
Similar results are obtained when investigating other pairs of historical
texts. I performed the computational experiment with the help of M. Za-
maletdinov (Faculty of Mechanics and Mathematics of Moscow University).
4,7.2. We now formulate another natural model of the evolution of written
information. If a historical text X describes events in a time interval (A,B),
then there is a correlation between the different informative functions con-
structed for X. For example, the graphs / f *(t,X) and fj{i } Y) for different i
and j attain local maxima at approximately the same points (years) in (A, B).
The K- model (statistical hypothesis) is subject to all the mathematical
corrections indicated above for the F-model; accordingly, we do not give them
here.
Its intuitive substantiation can be extracted from the following observation.
Say, if a year t from a time interval (A, B) is described in a historical text X in
more detail than the neighbouring years, then this circumstance must lead not
only to a local increase of the number of pages in X , which will have bearing
§4
Information Density in Texts
51
on the function fa, but also to an increase of references to the year t (this has
bearing on the function / 2 ), while the number of names of historical characters
mentioned at the same time in X increases locally, too (which has bearing on
the function fa), and so on. Roughly speaking, the longer a textual fragment
X(t ), the more names are mentioned there. Certainly, all these statements
(models) are valid only “on the average” and for large time intervals.
It is easy to see that the A-model is, actually, a corollary to the same basic
statistical hypothesis (information density conservation law) formulated in
Section 4.3 and is modelling the mechanism of loss and the fall into oblivion
of written evidence. To derive the K - model from it, it suffices to apply the
hypothesis to an arbitrary pair of informative functions /» and fj (and not
only to the volume function f\ = vol). In this sense, both the F- and the
K - model can be regarded as corollaries to the basic hypothesis of 4.3.1.
4.7.3. We now turn to the verification of the I< -model against concrete
historical data. It will be confirmed if the vectors a(fi(X)) and a(fj(X))
are proximal for most sufficiently large concrete texts X , i.e., if the number
d(fi(X),fj(X)) is sufficiently small. Omitting the diagrams, I shall give just
one typical example.
Sergeev’s monograph Essays on the History of Ancient Rome was again
taken as a text X for which the volume function vol X(t) and the frequency
of mentioning particular dates (years) fa(t,X) have been calculated. It turns
out that these two graphs are strongly correlated, i.e., their peaks practically
coincide. The calculations have shown that A = 25, p = 11, and B — A = 280
years (A = 241 B.C., and B = 521 B.C.). Finally, d(vol X{t),f 2 {t, X)) =
7 • 10“ 5 . Similar results have been obtained for other chronicles, textbooks,
monographs, and so forth, which we investigated. This confirms the A'-model.
The values of the coefficients obtained in the experiment are close to those
obtained by us earlier in verifying the F-model against necessarily dependent
pairs of historical texts.
4.7.4. We should not think that the coefficient d(fi(X),fj(Y)) is generally
small for the arbitrary texts X, Y, and the functions /,*, fj. If it were so, then
the extreme smallness of the numbers d obtained above (for texts known a
priori to be dependent) would not reveal anything.
Recall that, with the assumed uniform distribution of the random vector on
a simplex, all points of this simplex are treated equally during the random walk
of the vector a(Y) (and for a fixed vector a(X)). Hence, when the vector a(Y)
is moving farther from a(X ), the ball D(X, A) enlarges, and the coefficient
d(X,Y) approaches unity (the ball occupies a greater and greater part of the
simplex).
Concrete computations were then carried out for the independent historical
texts X and Y, i.e., texts describing events or periods of history (A,B) and
(C, D ) known a priori to be different. It turned out that the coefficient d was
of the order of unity for a small number of local maxima. For a large number
of maxima, this lower bound of the coefficient values for independent texts
52
New Statistical Methods for Dating
Chapter 2
decreases but continues to be several orders greater than its upper bound for
a priori dependent texts.
We illustrate this with a typical example. Consider the first and second
halves of Sergeev’s text as X and Y . They describe different historical epochs,
namely, the periods (A>B) from 521 to 381 B.C., and (C, D) from 381 to
241 B.c. in the history of ancient Rome. The functions f 2 (t,X) and / 2 (t, Y)
represent the frequencies of mentioning particular dates (years) in the texts.
It then turns out that A = 59 } B — A = D — C = 140 years, and p = 5.
Finally, d = 1/3, which is close to unity in contrast to the above examples of
dependent texts.
4.7.5. Our computational experiments confirm both of the theoretical hy-
potheses, i.e., the F- and K- models. We thereby discover specific laws of in-
formation density conservation in historical texts describing sufficiently large
time intervals. They are partly manifest in certain quantitative characteris-
tics of textual informative functions such as the distribution of local maxima.
Though their intuitive substantiation seems to be rather clear (see above), the
number of parameters of the informative functions, which can play the role of
“historical epoch invariants”, does not at all include all of them. For exam-
ple, the absolute amplitudes of the volume graphs, the frequencies of names,
and so forth, can be substantially different even for a priori dependent texts.
The amplitude is, therefore, not an invariant of the period. The discovery of
other “dependent text invariants” is a nontrivial problem and requires more
statistical research.
4.8. A new method for dating historical events. The method of
restoring the graph of the primary and surviving information
stock
4.8.1. We now offer a new method for dating historical events described
in ancient texts. The laws of information density conservation permit us to
introduce a formal procedure to date the events described in texts with lost
or unknown dating.
In fact, let Y be a historical text with undated events. Let it be supplied
with dates according to some unknown chronology. For example, let the years
t be counted from the unknown absolute date of the foundation of some city.
We can assume, nevertheless, that the text Y is parametrized by a time t in
the period between the years C and D according to the unknown chronology.
How can we restore the absolute dates of the events?
Construct the informative functions f%(t,Y) and consider the set of all ab-
solutely dated texts X. We also construct their informative functions fi(t>X)
and assume that we can choose X with some /,•($, X), or at once their whole
set, close in the sense of smallness of the coefficient d to an informative func-
tion /,(<, Y). In other words, d(fi(X),fi(Y)) is “small” (i.e., it is close to the
values of the coefficient d(/,(Z),/,(V r )) for surely dependent pairs Z and V).
It then means that within the framework of the F- or K- model substantiated
by us, the texts X and Y may be dependent. In particular, the time interval
§4
Information Density in Texts
53
(C, D) is close to ( A , B) described in X or simply coincides with ( A>B ).
Moreover, the smallness of d will indicate not only a possible proximity
of the time intervals (A,B) and (C, D), but also that of the events and the
historical epochs described (and even the coincidence). Meanwhile, it is im-
portant to bear in mind that the descriptions of the same events in X and Y
can be outwardly different, with different names or nicknames of the historical
characters, different geographical names, and so on. For example, the texts X
and Y can be two versions of the description (chronicle) of the history of a
region or state, but written in different languages, by different chroniclers, in
different countries, or according to different chronologies.
4.8.2. Let X\ y . . . be the collection of certain historical texts describing
events in the same time interval ( A y B ). Consider the averaged function of
their volumes, i.e., K(t) = vol(f;Xi,... y Xk) = £ volXj(f).
If the quantity of texts (i.e., the number fc) is sufficiently large, then the
graph of K(t) can be assumed, due to the computational experiments de-
scribed above, to be coincident with (or close to) the graph of the original
or surviving information stock C(t) or K(t), respectively. Meanwhile, the lo-
cal maxima of the graph of K(t) indicate those years in the time interval
( A , B) for which an especially large amount of written information (texts)
has survived.
This circumstance can help in dating available ancient historical texts.
Namely, especially extensive texts (describing individual years) must concen-
trate close to the local maximum points of the graph of K{i). To avoid a
misunderstanding, we stress once again that dating a text here implies dating
the events described in it. Besides, the text itself can have been written quite
recently, e.g., a textbook on ancient history.
4.9. The discovery of dependent (parallel) historical epochs
traditionally regarded as different
4.9.1. From 1978 to 1979, 1 made a series of experiments for calculating the
coefficients d(fi(X) 1 /,(Y)) for different pairs of historical surveys X and Y
which embrace considerable historical periods (A, B) and (C, D) of the same
length, i.e., B— A = D—C. This condition is necessary for a formal comparison
of the informative functions of different texts.
Unexpectedly, pairs of historical epochs (A,B) and (C, D) traditionally
regarded as different (and pairs of corresponding texts X and Y which describe
them) were discovered for which the coefficient d turned out to be extremely
small, i.e., characteristic of a priori dependent texts (epochs). Let me give
one example.
Let Z be the part of F. Gregorovius’ work History of the City of Rome in
the Middle Ages [3] embracing the events in medieval Rome from a.d. 300 to
754. In other words, A = 300, and B = 754. As a text Y, we take the part
of Livy’s History of Rome ([2]) describing the antique history of Rome from
the year 1 since the foundation of the City (Rome) to the year 459 since the
54
New Statistical Methods for Dating
Chapter 2
foundation of the City. It is traditionally assumed that the year 1 since the
foundation of the City coincides with 753 B.c. Therefore, we can take the 460-
year-long interval from 753 to 294 B.c. as (C, D). Traditional history certainly
regards these two texts, and the events described by them, as independent in
all respects. However, the computation shows that d(Z,Y) = 6 • 10~ 10 for the
volume function, which means that, due to the smallness of the coefficient,
the two texts and epochs are “like” dependent texts or epochs.
Recall that Sergeev’s text X [1] is necessarily dependent on Livy’s text Y [2].
This circumstance was confirmed above by a calculation according to which
d(X } Y) = 2 • 10“ 12 . We should, therefore, expect that from the standpoint of
the coefficient d, Gregorovius’ and Sergeev’s texts, i.e., Z and X, respectively,
will turn out to be dependent. Computations fully confirm this assumption,
too.
4.9.2. It iis interesting to investigate the behaviour of d by enlarging the
textual time scale. For example, let the parameter t now range not over years,
but over separate half-centuries. As a text Z , we take the part of Gregorovius’
book [3] which describes medieval Rome from A.D. 300 to 950, i.e., during a
650-year-long time interval. As a second text, A, we take the part of Sergeev’s
book [1] describing the events in ancient Rome from the year 1 to 650 since
the foundation of the City, i.e., from 753 to 103 B.c. (assuming that the year
1 since the foundation of the City coincides with 753 B.C.). Ranging over half-
centuries, the parameter f, therefore, assumes 13 values in the indicated time
interval. Calculations show that d(Z,X) = 1/50. Here, we have made use of
the textual volume graphs.
Similar results are valid for the information function fo both for the initial
(where t ranges over separate years) and the enlarged time scale.
The order of the coefficient d remains practically unaltered also in smooth-
ing the graphs of the functions f\ and / 2 , which indicates the stability of
the results relative to the smoothing, or averaging operation. We can also
see here the advantages of “long time scales”, when the length of the time
interval described in the texts is of the order of 100-1,000 years. With the
functions fi(X) and fi(Y) being correlated, the coefficient d then turns out
to be especially small for dependent texts. Enlarging the scale certainly makes
the picture rougher, which affects the increase of d. In the example given, it
increased to 1/50.
4.10. The dynasty of rulers and the durations of their reigns as an
important informative function
We now give an example of another important textual informative func-
tion. Consider a continuous (or gap-free) sequence R of some rulers (kings)
R \ y . .. ,R n • Let them be indexed by an integral parameter i whose increase
is associated with ordering the kings chronologically. As the informative func-
tion, we take the duration of a rule, i.e., let f&(i) = the duration of the rule of
§4
Information Density in Texts
55
king Ri who is ith in order. The sequence (dynasty) of the rulers J?i, . . . ) R n
will be called a dynastic stream for short <
Along with a sequence of numbers representing the durations of the rules,
we can define (as above) the vector a(R) of the local maximum points of this
graph, which permits us to compare two sequences of rulers by comparing the
graphs of the durations of their reigns.
The difference from the previous algorithm lies in the fact that the proximity
coefficient d must be calculable in a more complicated way for two dynasties,
which is related to the impossibility of regarding the random vector which
schematically represents the graph of the rule durations in a dynasty to be
distributed uniformly. Because of lack of space, we omit the details of this
investigation. It turns out that the distribution of the durations of the reigns
of kings (monarchs) is subject to a sufficiently nontrivial law which forms the
basis for defining the proximity coefficient for two dynasties.
4.11. Frequency distribution of the rules of kings who lived from
A.D. 1400 to 1800 and from 3000 B.C. to A.D. 1800
Figure 12 represents the result of processing the numerical information re-
garding the durations of reigns contained in J. Blair’s chronological tables [4].
The rule durations of historical characters of Europe and of the Mediterranean
region are marked off on the horizontal axis S in the time interval from 3000
B.C. to A.D. 1800, where the parameter 5 ranges not over separate years of a
reign, but pairs of years, i.e., 1-2 years, 3-4 years, 5-6 years, . . . , 89-90 years.
Further, the values of the following function, P(S) = number of historical
characters whose rule duration lies in the interval 5, are marked off on the
vertical axis P, For example, there were 55 personages (included in Blair’s
tables) ruling from 19 to 20 years. The dotted line in Fig. 12 represents the
frequency distribution of the rules of those kings who lived from A.D. 1400
to 1800. The continuous line describes those kings who lived from 3000 B.c.
to a.d. 1800 according to traditional chronology. In the investigation of the
complete lists of the European and Mediterranean rulers from 3000 B.C. to
A.D. 1800, I was assisted by M. Zamaletdinov and P. Puchkov.
The proximity coefficient d for two dynasties should be calculated by taking
into account the above histogram of the rule duration frequency. The follow-
ing experiment also indicates the necessity of resorting to it. All 1200 rulers
listed in Blair’s chronological tables were aligned in sequence, ordered chrono-
logically inside one dynasty (in one region). Simultaneous reigns were placed
in line one after another. We now index the obtained sequence of rulers by
i = 1,2, ... , 1200. Let & be the random variable representing the duration
of the ith king’s rule. Consider another random variable T)i(k) = £,•+*. The
sequence T] is thus obtained from f by shifting as a rigid block through k units
(numbers). Let r(k) be the correlation coefficient for £ and i}(k). The graph
of the variable r(k) for 1 < k < 300 is shown in Fig. 13. The calculations have
been performed on a computer by P. Puchkov. We do not have the space here
56
New Statistical Methods for Dating
Chapter 2
Figure 12. Frequency distribution of the rules of kings of Europe and the
Mediterranean region from 3000 B.C. to A.D. 1800 and from A.D. 1400 to
1800
to list the results of other experiments or the related conclusions. That will
be carried out in Vol. 2 of this book.
§4 Information Density in Texts 57
4.12. The concept of statistically parallel historical texts and epochs
We now formulate the concept of formal, statistical isomorphism (parallel) of
historical texts and epochs. Let a time interval ( A , B) in the history of some
region or state M be described in the historical texts X = {X\ , . . . , X*}. Let
another time interval (C, D) in the history of another and, in general, different
region or state H be described in the texts Y = {Yi, . . . , Y p }. Consider the set
of informative functions fi(X) and /*(Y) of these collections of texts. We will
say that the above historical epochs (texts) are formally isomorphic (parallel)
if the proximity coefficients d(/j(X), fi(Y)) are “small”. More precisely, they
must be as small as those of a priori dependent texts (epochs).
The concept of formal isomorphism of epochs described in some historical
texts does not at all mean that the epochs or events themselves are identical.
Their isomorphism (parallelism) can indicate the convention of ascribing, for
some reason, certain historical documents from one epoch to another. Our
goal is to solve the global problem, i.e., to describe the collection of all parallel
epochs and texts in the whole historical period of written language (see below)
for which we extend the stock of informative functions.
4.13. The “written biography” or enquete-code of a historical
character
4.13.1. Very heterogeneous historical information has survived concerning
the ancient dynasties of rulers. Meanwhile, different sources speaking of the
same ruler can be very different in the details concerning the description of
his or her activity. Sources can deal with the events during the monarches rule
differently, characterize differently the rulers themselves, and refer to them by
different names or nicknames, and so on.
But there exist more or less “invariant” facts whose description is less depen-
dent on a bias or political pressure on the chroniclers. One of these “invariant”
parameters, for example, is the duration of a king’s rule. Usually, there are no
special reasons for which a chronicler would like to considerably distort this
value (since it is emotionally neutral). By a “dynasty” of kings, we will under-
stand a continuous (i.e., gap-free) sequence of rulers of one region. We do not
assume that the throne should always be passed on hereditarily (from father
to son, etc.). With each ruler or prominent statesman playing an important
role in a particular period of history, we associate a certain table called an
enquete-code, or “formal biography”. By a “written biography” of a historical
character, we will understand the collection of all preserved evidence about
him or her. Normally, this “biography” is a set of individual facts which are
rather uncoordinated and traditionally ascribed to the “ruler” by the later
historians and chronologists based on the systematization and dating of the
available written evidence. Besides, arising from the investigation of the pri-
mary sources, this “biography” can have almost nothing to do with the actual
biography of the ruler. Strictly speaking, in many cases, we can only guess at
58
New Statistical Methods for Dating
Chapter 2
the ruler’s real biography. Therefore, we only deal with “written biographies”
in the following. Now, we associate each “written biography” established in
the primary sources with the enquete-code, or “formal biography” , by possibly
distinguishing all the basic facts from the “written biography”. In doing so,
we hierarchically order the facts of the “formal biography” in accordance with
their decreasing invariance. The facts most often distorted by chroniclers will
be nearer the end of the table.
(1) Sex of a personage: (a) male, (b) female.
(2) Length of life of the personage (or at least the year of death).
(3) Duration of the rule. It should be noted here that the end of the rule
is nearly always fixed uniquely by the chronicles. It is usually the death of
the ruler. The beginning of the rule sometimes (though, rather rarely) admits
several versions, among which are the official coronation date, those of confer-
ring the title of “Caesar” or “Augustus” , and of the death of a more powerful
co-ruler, and so on.
During the statistical investigation, all possible versions of the beginning of
a rule are then indicated by all available means, considered to be equally likely,
and included in the enquete-code. Note that, in analyzing real chronicles, we
found that the number of versions of the date of the beginning of a rule only
rarely reached three.
(4) Social status: (a) emperor, king, queen, etc., (b) army commander, (c)
politician, public figure, or statesman, (d) scientist, (e) religious leader (pope,
bishop, high priest, prophet, etc.).
(5) Cause of death of the personage: (a) natural, (b) on the battlefield, or
as the result of a mortal wound, (c) result of conspiracy in peace time, (d)
result of conspiracy in war time, (e) due to some special, exotic circumstance.
(6) Natural disasters during the personage’s rule: (a) hunger, (b) floods, (c)
epidemic diseases, (d) earthquakes, (e) volcanic eruptions. The duration of
the event and the year (or years) when it occurred are noted by all available
means.
(7) Astronomical phenomena during the rule: (a) did occur, (b) did not
occur, (c) solar or lunar eclipses, (d) appearance of comets (often as “swords”
in the sky, etc.), (e) “star” flares, (f) horoscopes, i.e., the planetary positions
relative to the zodiacal constellations.
(8) Wars during the rule of the personage: (a) did take place, (b) did not
take place.
(9) Number of wars (different wars are usually separated in chronicles).
(10) Basic time characteristics of the wars B\ 9 B 2 ,." y B p . Namely, a* =
year of the king’s rule in which the war numbered fc, i.e., Bk , took place; h* =
duration of the war J3*; Ck x = distance in years between the wars B * and B x .
(11) Intensity of the war Bk for each k. The intensity of a war can be
estimated, for example, by the volume of texts in chronicles devoted to it.
Roughly speaking, wars can be divided into two classes, namely, (a) large-
scale wars, (b) local wars.
(12) Allies, adversaries, and neutral forces in a war Bk (for each k); their
§4 Information Density in Texts 59
number and schematic diagram of their relations (who are allies or enemies,
etc.).
(13) Geographical localization of a war Bk (for each k): (a) near or inside the
capital, (b) inside the state, (c) outside the state and where exactly (external
war), (d) both external and internal war simultaneously, (e) civil war or a war
with external enemy.
(14) Final result of the war: (a) victory, (b) defeat, (c) uncertain result.
(15) Peace talks: (a) concluding a peace treaty after the victory of one of the
adversaries (who exactly; see paragraph (12)), (c) concluding a peace treaty
after the defeat of the ruler.
(16) Conquering the capital: (a) did occur, (b) did not occur, (c) specific
circumstances of the capital’s siege or its fall.
(17) Fate of the peace treaty: (a) it was violated (by whom and under what
circumstances), (b) was not violated during the rule.
(18) Detailed description of conquering (or fall of) the capital during the
war.
(19) Diagram of the armies’ marches during the war.
(20) Ruler’s participation in the war: (a) did occur, (b) did not occur.
(21) Conspiracies during the ruler’s lifetime: (a) did occur, (b) did not occur.
(22) Geographical localization of wars, allies, adversaries.
(23) Name of the capital. A translation of the name is necessary.
(24) Name of the state. Translation is necessary.
(25) Geographical localization of the capital (with the terms translated).
(26) Geographical localization of the state (with the terms translated).
(27) Legislative activities of the ruler (a) reforms and their nature, (b)
issuing of new code of laws, (c) reintroduction of former laws (which exactly).
(28) Complete list of all names of the ruler with their translations. As a
matter of fact, practically all ancient names have meaningful translations and
originally were simply nicknames (such as “mighty”, etc.).
(29) Ethnic group of the ruler, members of his or her family, composition
of the family.
(30) Ethnic group of people living in the region or city.
(31) Founding of new cities, capitals, fortresses, harbours, etc.
(32) Religious situation: (a) introduction of a new religion, (b) sectarian
struggle (between what sects exactly, names of the leaders and their transla-
tions), (c) religious riots and wars, (d) religious meetings, councils, etc.
(33) Dynastic struggle inside the ruler’s clan, murders of relatives (if any),
usurpers of the throne, adversaries, etc.
(34) Other fragments of the “personage’s biography” will not be differenti-
ated in such a detailed manner and will be collected in this item. We will call
the information gathered here the “biographical remainder” . It is convenient
to measure it as a percentage of the whole “biography” .
Denote the listed items by EC-1, EC-2, ... , EC-34 (enquete-code, para-
graph (1), (2), . . . , etc.). The whole enquete-code (i.e., the above table) will be
denoted by EC for short. Thus, each “written biography” can be represented
60
New Statistical Methods for Dating
Chapter 2
as a certain formal table of EC, or “formal biography”. Certain items in this
table may be left blank, which occurs in the case where the corresponding
information has been lost in the surviving documents.
4.13.2. If the parameter i ranges over the numbers of consecutive rulers in a
dynasty (dynastic stream) J?i, . . . y R ni then the enquete-codes EC, of Ri can
be regarded as the set of values of a certain new informative function. Since
the set of enquete-codes of the rulers of a dynasty practically completely ac-
cumulates all the important information from that epoch, we can assume that
the sequence, in fact, describes the epoch ( A y B ) in the history of the region.
It is “covered” by the rulers of the given dynasty. Eventually, we can asso-
ciate each historic epoch with a set of informative functions v = {/i,/ 2 > • • • ,
EC-dy nasties), where /i,/ 2 , • . are the informative functions already famil-
iar to us, and the EC-dynasties are the enquete-code collection for the rulers
“covering” ( A y B ). Since the rulers in one region sometimes reign simulta-
neously (and are then called co-rulers), different “dynastic jets” should be
distinguished from the total dynastic stream, i.e., continuous (gap-free) sub-
sequences of personages among whom there are no co-rulers or very few of
them.
Consider now the two epochs (A, 5) and ((7, D). Associate each with the
above set of informative functions, i.e., v = {/i,/ 2 , - * , EC-dynasties} and
v ' = {f{y / 2 > • • • > EC'-dynasties}.
4.14. A method of comparing the sets of informative functions for
two historical epochs
4.14.1. Let us describe a method of comparing the sets of informative func-
tions v and v' for two epochs. The comparison has been carried out in terms
of coordinates, i.e., the functions /,• and // of the same kind are compared
to each other. For /i, / 2 , . . . , the coefficients of the above d- type were calcu-
lated. The comparison of the dynastic enquete-codes, i.e., the EC-dynasties
and EC'-dynasties, is more complicated owing to a finer structure of these
informative functions. Without going into details, we only discuss the com-
parison principle.
We start with the comparison of rule durations. Let T \ , . . . , T n and T [, . . . ,
T„ be two sequences of dynastic rule durations in the epochs (A, B) and
(C, D), respectively. We will measure the “distance” between them using
the coefficient A introduced in §5; see below. The coefficient A can be called
“stream deviation coefficient” (SDC).
4.14.2. Thus, we compared the items of the enquete-codes EC-3 and EC'-3
of two rulers R and R'. The comparison of the remaining items will be done
as follows. Let EC-1, . . . , EC-34 be the enquete-code EC for the ruler R, and
EC'-l, . . . , EC'-34 the enquete-code EC' for R f . We introduce the numerical
coefficients E \ , . . . , £34 measuring the proximity (or remoteness) of the items
EC-1 and EC'-l, ..., EC-34 and EC'-34. We have omitted here the times
EC-3 and EC'-3, since they are compared already by means of the SDC. To
§4
Information Density in Texts
61
compare the items EC -j and EC ; -j, we introduce the coefficients E-j. Then
the following three situations are possible.
(1) The biographical data compared are similar. For example, the items
EC-5 and EC'-5 state that both rulers died a natural death, in which case we
put E - 5 = +1.
(2) The biographical data compared are not explicitly coincident. For ex-
ample, EC-5 states that a ruler died a natural death, and EC'-5 that he died
as the result of a conspiracy. We then put E-b = — 1.
(3) The biographical data compared are neutral in the sense that they are
consistent, but not identically coincident. For example, EC-5 states that a
ruler “died”, whereas EC'-5 states that a ruler “was killed”. We then put
E-b = 0.
We now introduce the resultant coefficient E = E-l + E-2 + E-4 + • • • +
E- 33. The coefficient E- 3 is absent because the rule durations are compared
by means of the SDC. Thus, the coefficient E measures the proximity of
the enquete-codes of the two rulers R and Rf. Now, given two sequences of
rulers {Ri} and {/ZJ}, we obtain a sequence of coefficients E( comparing them.
Finally, we introduce the average coefficient J 5 av . = £ measuring
the proximity of the complete enquete-codes of the dynasties of {Ri} and
{iZ(*}. Collecting this information, we might be able to estimate the remoteness
or proximity of the enquete-codes of two dynasties and their corresponding
epochs by means of the coefficients £? av . and the SDC.
4.15. A computational experiment
From 1978 to 1979, I performed a computational experiment, comparing sev-
eral hundred pairs of epochs (A, B) and (C,D), i.e., sets of their informative
functions v = {/i, / 2 , . . . , EC of dynasties of rulers}. The time limits for the
experiment were 3000 B.c. and A.D. 1800, with the events localized in Europe,
the Mediterranean region, Egypt, and the Near East.
In particular, the experiment showed that, in comparing the enquete-codes
of some personages, we often had to put the coefficient E-j equal to zero, which
occurred because the information compared there was consistent and, at the
same time, unconfirmed. The role of +1 and —1 was therefore heightened. We
discovered further that, in the overwhelming majority of concrete enquete-
codes, the coefficient E - 34 has to be made equal to zero, again because of the
consistency of information compared. Recall that item EC-34 of an enquete-
code is the personage’s “bibliographical remainder”. For a reliable comparison
of the items EC-34 and EC'-34 in two biographies, we must be certain that
we really possess sufficiently complete “written biographies” of the personages
compared. However, to guarantee the completeness of the rulers’ enquete-
codes (all the more, of whole dynasties) is usually a complicated matter, due
to which we have resorted to the following formal method.
For each historical epoch, we have chosen a (possibly) uniform and suffi-
ciently large historical text describing the events. We took either a fundamen-
tal monograph of the type of Gregorovius’ work or a fundamental primary
62
New Statistical Methods for Dating
Chapter 2
source of Livy’s type. Then item EC-34 of the enquete-code was identified
with the “bibliographical remainder” for a historical character given in the
historical work. For simplicity, the volume of item EC-34 was usually calcu-
lated as a percentage of the volume of the entire “biography” of the personage.
Meanwhile, as a rule, we did not analyze the components of item EC-34.
We illustrate this with an example. The epoch of regal Rome from 753 to 510
B.c. described by Livy (see [2]) turns out to be sufficiently close to that of the
Roman Empire in a.d. 300-552, with the SDC being less than the lower bound
for most of the SDC values for independent dynasties. Further, it turns out
for these epochs that E * v . = +19, and that the volume of item EC-34 equals
29%. In other words, about 29% of the “biographies” of historical characters
in these epochs were “thrown overboard” from the discussed parallels, it being
important that this entire “biographical remainder” is zero for the coefficient
of EC-34, which means that the information not involved in the parallel is
non contradictory within the framework of the comparison.
It is useful to compare this result, which probably indicates the dependence
of the epochs, with the numerical parameters found for an arbitrarily chosen
pair of independent epochs. As an example of independent historical epochs,
we take the following dynasties.
(1) The dynasty of the Russian grand dukes from Igor (912-944) to
Demetrius I (1275-1293).
(2) The dynasty of the Byzantine emperors from Theodosius II (408-450)
to Theophilus (829-842) [4]. Calculations show that the SDC is “very large”;
further J? av . = —8.7. In this case, the volume of the “remainder” EC-34 is
equal to 40%. This pair of dynasties is independent from the standpoint of
the coefficients we introduced.
A great difference is obvious between this and the example of the two de-
pendent Roman Empires (see above).
We shall now describe the results of the global computational experiment
of comparing different epochs.
4.16. The remarkable decomposition of the global chronological
diagram into the sum of four practically indistinguishable
chronicles
4.16.1. The application of the above methods to the material of the GCD
led to the discovery of isomorphic, parallel historical epochs in the history of
Europe and the Mediterranean region (see Fig. 14). The sequence of figures
forms a line in the diagram, schematically representing the history of ancient
and medieval Europe. Identical geometric figures (denoted by the same letters)
schematically represent historical epochs (or some parts of these epochs) which
turn out to be formally isomorphic and parallel.
Figure 15(1) represents the decomposition of the GCD into the sum of
four practically indistinguishable chronicles. We can say that the GCD is
decomposable into the sum of several shifts of the same chronicle. As it turns
out, we can distinguish a shorter part in the GCD, which we call the chronicle
§4
Information Density in Texts
63
Figure H. Parallel historical epochs in the history of Europe (and the Mediter-
ranean region) and in biblical history.
C\. We then take another three copies of the same chronicle, each of which
is shifted back on the time axis (i.e., from right to left) by 333 years, 1,053
years, and 1,778 years, respectively, after which all the shifted copies of C\ are
glued to it and to each other, resulting in a longer GCD chronicle on which,
therefore, parallel epochs duplicating each other appear.
4.16.2. We now give a short description of the GCD for Greece, Rome, and
Germany. We shall move from left to right in Fig. 14 and list the historical
epochs denoted by different geometric symbols. For a detailed description of
all the dynastic jets for these cases, see Vol. 2 of this book.
The numbers of the following items correspond to those of the indicated
historical epochs (from 1 to 15).
(1) The Trojan kingdom of 1460-1236 B.c. Seven legendary Trojan kings.
(2) The TVojan War, ca. 1236-1226 B.c. The fall of Troy, expulsion of the
TVojans.
(3) Several dynastic jets of ancient Greek rulers from 1226 to 850 B.C., i.e.,
from the fall of Troy to the second version of dating the TYojan War according
to the ancient authors Hellanic, Damast, and then Aristotle ([5]; [5*], p. 23)
immediately before the foundation of Rome.
-1700
64
New Statistical Methods for Dating
Chapter 2
S 8 > H 88 § ?
Original Co: K
C,: K
333-year shift of C, K
P 111 C
Sewing C^C,: K I II K
T T
1 .053-year shift of C 1
T T
Sewing C^C^C,: K 1/ H \l K
1.778-year shift of C,
T T T T
P C
C,+C, +Cj
=GCD
l p ll c ll p ll c l
-1700 -1300
w z M
Byzantine history QH w i 1 i w I
TT T T T
•900 -500 -100 +100 +500 +900 +1300 +1700
Figure 15(1). The remarkable decomposition of the GCD into the sum of four
short and practically indistinguishable chronicles.
§4
Information Density in Texts
65
(4) The foundation of Rome and the regal period described by Livy, from
760 or 753 B.c. to 522 B.c.
(5) The war with the Tarquins, the kings’ “exile” from Rome and the foun-
dation of the ancient Roman republic (522-509 B.C.).
(6) Republican Rome and ancient Greece in 509-82 B.c. The end of classical
Greece, start of Hellenism.
(7) Civil wars in Italy during the fall of republican Rome in the 1st century
B.c. Beginning of imperial Rome. The Roman Empire (82 b.c.-a.d. 217).
(8) Wars in Italy and crisis of the Roman Empire in the middle of the 3rd
century A.D. Wars with the Goths, “soldiers’ Emperors” (a.d. 217-235-251).
(9) Restoration of the Roman Empire under Aurelian, and contemporary
war in Italy (a.d. 270-300).
(10) Roman Empire a.d. 300-535. Western and Eastern Empires.
(11) Gothic War in the middle of the 6th century a.d. in Italy (535-552).
(12) Medieval papal Rome from A.D. 553 to the middle of the 10th century.
(13) Carolingian Empire, including the empire of Charlemagne 681-887.
Wars of Charlemagne.
(14) Holy Roman Empire of the German Nation in the 10th to 13th century.
The war in the middle of the 13th century; the fall of the dynasty of the
Hohenstaufen (1250-1268).
(15) Empire of the Habsburgs (1273-1619 or 1637).
Besides, (10)— (13) also include the medieval dynastic branches of the East-
ern Roman and Byzantine Empire.
The parallels marked in Fig. 14 sometimes link all duplicate historical
epochs, and sometimes only certain layers within these epochs. Certain epochs
in the GCD can branch into several layers parallel to other epochs.
4.16.3. Consider the epoch of ancient Rome from 753 b.c. to 230 B.c. and
that of medieval Rome in from 300 to 820. Remember that these epochs
are “parallel” in the sense that the coefficient d measuring the proximity of
the local maxima of the volume function (for the primary sources describing
these periods of history) is very small and equals 6 • 10“ n . This parallelism
(overlapping) is confirmed by that of the enquete-codes which I found for
the rulers’ dynasties of these matching epochs. Moreover, I have discovered
the parallelism of the events of the epoch of ancient Rome from 753 B.c.
to a.d. 300 and that of medieval Rome from A.D. 300 to ca. 1353, which
follows. The overlapping of parallel events occurs in shifting their dates by
ca. 1,053 years. In other words, this rigid chronological shift can be written
as the formula T = X + 300 years, where T are years A.D., and X the years
from the foundation of the City (Rome). It is assumed traditionally that the
year 1 since the foundation of Rome coincides with 753 B.c. In our forward
shift, the “foundation of the city” falls in the year A.D. 300. I discovered this
important “uniform forward shift” formula as a result of applying the enquete-
code method and the method for the calculation of d. It turns out that J5 av . =
-f 18 in the time interval from the year 1 to 250 since the foundation of Rome
(when compared with the duplicate period A.D. 300-550). In the next time
66
New Statistical Methods for Dating
Chapter 2
interval, a.d. 550-820, the small value of the coefficient d calculated above for
the duplicate epoch 250-520 since the foundation of Rome is consistent with
the existence of a whole series of far-reaching parallelisms linking these two
periods, i.e., the antique and medieval ones. An additional analysis of this
overlapping (in the interval from a.d. 553 to 820) was then carried out by
E.M. Nikishin. From 1978 to 1979, I also investigated the next time interval
from the middle of the 9th to the 17th century (which overlaps with the period
from 200 B.c. to A.D. 570) with the aid of the enquete-code method. The result
is shown in the GCD in Fig. 14. For details, see Vol. 2.
4.16.4. As noted above, the concept of a text with a scale is more general
than the examples of historical, narrative texts given above. For example,
as a text A, we can take the collection of all the works of one author, as
the parameter the numbers of pages (with consecutive pagination), and some
quantitative characteristic of the text, for example, the average length of sen-
tences, frequency of conjunctions, and so forth, as the informative function.
The question arises whether there exist any conservation laws controlling the
behaviour of such informative functions. It turns out that the answer is pos-
itive (see the author’s paper “Authorial invariants in Russian literary texts
of narrative sources” in Methods of Quantitative Analysis of Texts of Nar-
rative Sources , History Institute of AN SSSR, Moscow, 1982, pp. 86-109; in
Russian).
We stress that the present chapter is only a brief survey of the theses,
the detailed treatment of each of which is rather voluminous and requires an
extensive machinery designed for lengthy statistical material.
References
[1] Sergeev, V.S., Essays on the History of Ancient Rome . Sotsekgiz, Moscow,
1938 (in Russian).
[2] Livy, Titus, Works. Cambridge, Harvard University Press; Heinemann,
London, 1914.
[3] Gregorovius, F., History of the City of Rome in the Middle Ages. G. Bell
& Sons, London, 1900-1909.
[4] Blair, J., Blair f s Chronological and Historical Tables , from the Creation
to the Present Time , etc. G. Bell & Sons, London, 1882.
[5] Niese, B. Grundrifi der romischen Geschichte nebst Quellenkunde , 2nd
ed. Miinchen, 1923.
[5*] Russian transl. (from the 1st ed.), St. Petersburg, 1908.
§5 Chronology of Ancient Dynasties 67
§5. A Method of Duplicate Recognition and Some Applications to
the Chronology of Ancient Dynasties 1
5.1. The process of measuring random variables
Let a finite set of points D and a certain many-valued mapping V : D — ► R n
which transforms D into a larger, but still finite set of points V(D) be given
in the Euclidean space R n . For example, V can model a multiple process of
measuring a certain random discrete variable £ taking values in the set D.
The many-valuedness can be caused by the nonuniqueness of the results of
measuring due to the existence of random errors. Meanwhile, V(D) can be
regarded as the set of values obtained by measuring the given random variable.
Note that each true value x of £ turns into the set of points (values) V(x)
upon its measuring. It represents the original value x of the variable £. In
particular, each point of V'(x) can be regarded as an approximate value of the
true value x.
In studying real processes of measuring random variables, the principal
difficulty lies in correctly modelling, by means of the choice of a suitable
mapping V , the mechanism of real measuring errors. Now, let the set D of the
real values be unknown, and only the set V(D) of the “results of measuring”
the desired random variable be known. How can we recognize those points of
V(D) which correspond to the same point in the set D? The points (results
of measuring) associated with the same real, true value will be called its
duplicates, which, in turn, can naturally be called “original”.
Let the mapping V be such that the sets V(x) and K(y) are disjoint if the
points x and y are different.
5.2. The distance between two random vectors
We introduce a certain natural measure A of the distance between the points
in the set V(D). We shall strive to make the points which belong to the same
set V*(x) (i.e., duplicates) sufficiently close in the sense of the measure A. On
the contrary, points from different V(x) and V(y) should be distant in the
sense of A.
Let a and b be two points in V(D). Fix a and construct its special neigh-
bourhood H r . We shall attempt to make the point a the centre of H r , and let
b lie on the boundary of the neighbourhood or close to it.
The simplest of such neighbourhoods is = {c € R n : |a, — c,*| < |a, —
6,* |, 1 < i < n}. In other words, H' r is simply a parallelepiped with its centre at
a and having b as one of its vertices, where a = (ax, . . . ,a n ), b = (&x, . . . , 6 n ).
For this simplest construction to become suitable in important applications,
we have to extend it to model the mechanism of the random errors of interest,
which influence the measurements of the true values of the variable £. We
1 First published as an article in DAN SSSR 258, 6(1981), pp. 1326-1333 (in Russian).
68
New Statistical Methods for Dating
Chapter 2
construct such a neighbourhood H r (a y b) below and thereby introduce the
natural measure A, permitting us to estimate the distance between two points
a and b. As the basis for defining the measure, we shall take the procedure
developed in §4. Namely,
AM)
vol# r (a,6)
volV(D)
where vol V(D) is the number of points in the whole set V(D ), and vol H r (a, b )
that from V(D) in the neighbourhood H r (a,b).
5.3. Dynasties of rulers. The real dynasty and the numerical
dynasty. Dependent and independent dynasties. The
small-distortion principle
We shall now describe a concrete problem for whose solution we introduce the
measure A. Let a historical text describing a previously unknown dynasty of
rulers be discovered with an indication of the duration of their reigns. The
question arises whether this historical dynasty is new and not mentioned in
the known documents or whether it is one of the rulers’ dynasties already
known to us but described in a text in unusual terms (with the rulers’ names
distorted, etc.).
Consider n consecutive authentic rulers (kings). Let the true rule durations
of these kings be pi,P2> • • • ,Pn, respectively. We call this sequence a real dy-
nasty. Note that the same real dynasty of rulers is often described in the
primary sources from different standpoints by different chroniclers. But there
exist more or less “invariant facts” concerning these rules, and their descrip-
tion depends little on the tastes of the author of a primary source (chronicler).
Such facts include, for example, the duration of a king’s rule, since there are
usually no special reasons for which the chronicler should considerably and
intentionally distort it. Nevertheless, chroniclers often encounter serious dif-
ficulties in calculating the regal rule duration, which leads to giving different
values to the duration of the rule of the same historical character in different
historical documents.
Thus, each author (chronicler), while describing a real dynasty p = (p\,P 2 ,
• • • >Pn)> calculates the duration a t of a king’s rule and obtains a certain se-
quence of numbers a = {a\ , « 2 , . . . ,a n ). This sequence of numbers represented
as an integral vector a in the space R n will be called a numerical dynasty. An-
other chronicler, while describing the same real dynasty of kings, will possibly
obtain another vector 6 from R n , i.e., another numerical dynasty. Thus, one
and the same real dynasty can be represented as different numerical dynasties
in different documents.
As the set D described in 5.1, we take a sufficiently large set of real dy-
nasties of length n, i.e., D = {p = (pi, . . . ,p n )}. We formulate the following
theoretical model (statistical hypothesis).
§5
Chronology of Ancient Dynasties
The small-distortion principle. If two numerical dynasties are sufficiently
close (in the sense of the measure A ), then they indeed represent the same real
dynasty of kings, i.e. f they are merely two different versions of its description.
Such numerical dynasties will be called dependent. On the contrary , if two
numerical dynasties represent two real dynasties of kings , known a priori as
different , then the numerical dynasties are much different from one another
(in the sense of the measure X). Such numerical dynasties will be called inde-
pendent.
Later in Section 5 . 4 , before verifying this model experimentally, we shall
give an exact description of the measure A. Meanwhile, we identify the set of
all numerical dynasties describing real historical dynasties from the set D in
the space R n (see above) with the set V(D).
5.4. Basic errors leading to controversy among chroniclers as to the
duration of kings 9 rules
We now point out concrete errors most often leading to the controversy among
chroniclers as to the duration of the rules of kings.
(a) Permutation of the names of (or confusion between) two neighbouring
rulers.
(b) Replacing two neighbouring rulers by one, the duration of whose rule
was assumed to be equal to the sum of the rule durations of both.
(c) Computational error by a chronicler. The longer the duration of a king’s
rule, the greater the error that arises in its computation.
It turns out that these three basic types of concrete errors made by chron-
iclers can be sufficiently simply described by means of a suitable mapping,
V : D — ► R n . Let p be a certain real dynasty in the set D. We call the dy-
nasty (vector) c a virtual variation (virtual vector) of the dynasty p and write
c = v(p) if the following conditions are fulfilled, namely, each coordinate c t
of c coincides with one of the three coordinates of the original vector p, i.e.,
Pi-i , Pi > Pi+i , or with pi + p <+ 1 .
It is clear that each of such virtual vectors, or virtual dynasties, can be
regarded as a numerical dynasty and be obtained from a real dynasty p,
because of chroniclers’ errors of type (a) and (b).
Eventually, we take as V(D) the union of all virtual vectors (virtual dynas-
ties) c = v(p ), where p ranges over all the real dynasties of D. It remains to
model an error of type (c).
Let a piecewise smooth, nonnegative function a(t) be given on the positive
half-axis t > 0 . In our case, the role of a(t) will be played by the probability
density of a random variable rj to be specified below. Put h(t) = /(a(t)),
where f(s) is a certain monotonically decreasing function of a parameter s,
given on the half-axis s > 0, and such that lim/(s) = -foo when s — ► 4-0.
For example, as /($), we can take the function j. If rj is a discrete random
variable with probability density a(tf), then the quantity h(t) becomes greater
as rj assumes the value t with lesser probability. In our problem, we take as 77
the duration of a king’s rule in a dynasty. Let t range over all positive integers
70
New Statistical Methods for Dating
Chapter 2
(i.e., possible values of the rule duration). If t is a certain fixed rule duration,
or the value of 17, then a(<) will mean the number of historical characters
ruling for t years (see Fig. 12). We call h(t) the error amplitude in measuring
a rule t years long. The graph a(t) in Fig. 12 shows that short rules are most
frequent, and, conversely, long rules are rare.
5.5. The experimental frequency histogram for the duration of the
rules of kings
§4 showed the experimental frequency histogram I obtained for the rule du-
rations of authentic kings (Fig. 12). If t is the value (rule duration) taken by
the random variable rj with large probability, then the amplitude of chroni-
clers’ errors h(t) decreases. In other words, the values of short rule durations
of frequently mentioned kings have been calculated by the chroniclers better
than long ones, which are rarely encountered. We now indicate the error func-
tion h(t) of chroniclers, which we calculated for the probability density of the
random variable, namely the “rule duration” (see §4).
Break the interval from zero to 100 on the integral axis t into smaller seg-
ments of the form (lOJfc, lOir -f 9), where k = 0, 1, . . . ,9. Then the amplitude
h(t) of the chroniclers’ error has the following form, namely,
h(t) = {
2
3
1 - 1 )
when 0 < t < 20
when 20 < t < 30
when 30 < t < 100.
5.6. Virtual dynasties and a mathematical model for errors made
by the chronicler in measuring the rule duration
Consider a rectangular parallelepiped II (a, b) in the space IP 1 and denote it
simply by II. Its orthogonal projections 7r, = a,' ± (|a,* — bi\ + h(a*)) onto the
coordinate axes in the space R n will be given by intervals with the following
endpoints, namely,
a,- ± (|aj - 6j | + 2)
a» ±(|a> ~ M + 3)
a,±(|a,-6,| + 5[fj]-5)
if 0 < a» < 20
if 20 < a { < 30
if 30 < a, < 100,
where [y] denotes the integral part of the number y. Thus, if 0 < a,- < 20, then
the rule duration a t , and also 6 t , of two kings numbered i in the dynasties
compared, is considered by us only approximately, to the accuracy of ±1 year.
In other words, this is an error of the chronicler, made in measuring the rule
duration. If 20 < a* < 30, then the chronicler’s error is already equal to ±1.5
years, and so forth. We now fix two dynasties a and 6.
It remains to model the fact that the assignment of a point (dynasty) c
from the set of virtual dynasties V(D) to the parallelepiped II can be consid-
ered only approximately, with some allowance. Hence, we have to make the
§5
Chronology of Ancient Dynasties
71
boundary of II less distinct. Let r be a certain fixed number. Consider a real
dynasty p from the set D. Assume that at least r coordinates p t * of this vector
p, i.e., r values for the rule duration, have fallen onto the projections ;r, of II.
We assume, in addition, that a certain virtual dynasty c = v(p) of p has fallen
entirely into II. We say that such vectors (dynasties) p from D are r-close to
the parallelepiped II determined by the two fixed dynasties a and 6.
Eventually, we define the neighbourhood H r (a i b) of a by considering the
union of II and all the virtual variations of dynasties (vectors) p from the set
£>, which happened to be r-close to II (see Fig. 15(2)).
Parallelepiped n(M,N)
Figure 15(2). Parallelepiped determined by the two fixed dynasties M and N.
As a proximity measure for two dynasties a and 6, we take the ratio of the
number of dynasties (vectors) of the set V(D ), which are in the neighbour-
hood H r (a,b ), to the total number of dynasties (vectors) in the set V(D).
In counting the number of dynasties in // r (a,6), we do not count the virtual
variations of a and 6, which are different from them.
The constructed number A has an important probabilistic interpretation.
Indeed, construct the function <f> of the probability density for the random
vector from the set V ( D ) of all virtual dynasties, with the vector ranging over
the dynasties. We divide the space R n into standard cubes of sufficiently small
size, so that no point of V(D) has fallen onto any of their boundaries. If x
is an interior point of some cube, then we take the following value as <t>(x),
namely,
<t>{x) =
number of points of the set V(D) in this cube
total number of points in the set V(D)
72
New Statistical Methods for Dating
Chapter 2
However, if x is on the boundary of the cube, then we put = 0.
It is clear that the above measure A(a,6) is the integral of <j> with respect
to the set # r (a,6), provided that //r(a,6) consists of the cubes of our parti-
tion. Since we model here approximate calculations of the chroniclers, we can
assume that this condition is fulfilled (if the cubes are sufficiently small).
Finally, the number A(a,6) can now be regarded as the probability of the
fact that the random vector distributed in the space iZ n with the density
function <f> has fallen into the neighbourhood H r with its centre at a point
(dynasty) a and the “radius” |6 — a\ + h(a).
5.7. The small-distortion principle and a computer experiment
To verify the theoretical model of Section 5.3 (the small distortion principle),
the chronological tables of J. Blair [1] and F. Ginzel [2] were used; they contain
practically all the basic chronological data that survived for real historical
dynasties. I have made a complete list of all dynasties of length n = 15
from the history of Europe, the Mediterranean region, the Near East, and
Egypt from 4000 B.C. to A.D. 1800. The data have then been supplemented
by information from 14 other chronological tables. The obtained list D turned
out to represent certain real kings by several different numerical dynasties
(due to the difference in the chroniclers’ opinion). We now indicate the basic
historical dynasties included in D.
Bishops and popes in Rome, Saracens, high priests in Judaea, Greeks in Bac-
tria, exarchs in Ravenna, all dynasties of Pharaohs and other Egyptian rulers,
dynasties of the Byzantine Empire, Roman Empire, Spain, Russia, France,
Italy, the Ottoman Empire, Scotland, Lacedaemon (Sparta), Germany, Swe-
den, Denmark, Israel, Babylonia, Syria, Sicyon, Judaea, Portugal, Parthia,
the Bosphorus, Macedonia, Poland, and England.
The total number of dynasties making up the virtual set V(D) in the space
R 15 turned out to be approximately equal to 15 • 10 11 . If, for some pair of
virtual dynasties a and 6, the number A(a,6) is sufficiently small, then the
observable proximity of the dynasties a and 6 is a rare event; the rarer it is,
the less is the coefficient A(a, 6). As r in the numerical experiment, we have
taken 11 equal to 1 + |n by the “two-thirds” rule.
The author then performed an extensive computational experiment to de-
termine A(a,6) for different pairs of dynasties a and b. The result fully con-
firmed the model of Section 5.3. Namely, the coefficient A (a, b) turned out to
oscillate for surely dependent numerical dynasties in the interval from 10” 12 to
10” 8 . If, on the contrary, the numerical dynasties a and b are surely indepen-
dent, then the coefficient A(a,6) was not less than 10” 3 . The great (5th-order)
difference between surely dependent and surely independent dynasties is man-
ifest. The remaining ones make up a small percentage of the total number of
dynasties.
Obviously, the above result permits us to solve the problem of distinguishing
dependent numerical dynasties.
§5 Chronology of Ancient Dynasties 73
5.8. Pairs of dependent historical dynasties previously regarded as
independent
Our experiment has discovered several special pairs of historical dynasties
a and b which previously had been regarded as independent in all senses;
however, the value of the coefficient A(a,6) for them is the same as for pairs
of surely dependent dynasties. There are only several dozen such special pairs
among the 10 6 of dynastic pairs studied.
We shall illustrate this with some examples.
(1) The Roman Empire from 82 B.c. to a.d. 217 and the Roman Empire
from 270 to 526, where A = 1.3 * 10” 12 .
(2) The Holy Roman Empire of the German Nation from 962 to 1254 and
the Habsburgs Empire from 1273 to 1619, where A = 1.2 • 10” 12 .
(3) The Roman Empire from 270 to 553 (see example 1) and the Holy
Roman Empire of the German Nation (see example 2) from 962 to 1254,
where A = 2.3 • 10” 10 .
(4) The Carolingians, the empire of Charlemagne from 681 to 887 and the
Eastern Roman Empire from 333 to 527, where A = 8.25 • 10” 9 .
5.9. The distribution of dependent dynasties in the “modern
textbook” of ancient history
All the above results have been analyzed in the following manner. I have
constructed the GCD for all the historical dynasties described in Section 5.7
(see also §4 and [3]), for which the rule duration periods for all the rulers of the
indicated dynasties from list D and the dates of all basic events occurring in
the time interval from 4000 B.c. to A.D. 1800 were marked off on the horizontal
time axis (as horizontal intervals of different length).
The GCD was then subjected to the procedure of discovering duplicates,
or dependent epochs. All the discovered historical dynastic pairs a and 6, and
the corresponding historical epochs, for which the coefficient A (a, 6) turned
out to be anomalously small, of the order from 10” 12 to 10” 8 , were marked
on the GCD. We will call such dynasties (and also the epochs) duplicates.
Recall that the theoretical model of Section 5.3 has been confirmed by the
results of the experiment performed, from which it follows that the anomalous
small value of A(a, 6) most probably indicates the dependence of the historical
dynasties and their corresponding epochs.
We now describe the portion E of the GCD in the time interval from 1600
B.c. to A.D. 1800. We represent the result as a schematic line E made up of
consecutive letters indicating on the time axis different dynasties and the cor-
responding epochs. Duplicate epochs will be represented by identical letters.
Because of the enormousness of the data, we give here only a rough sketch of
the GCD (for details, see Vol. 2 of this book). The letters in the numerator
and denominator of a fraction represent simultaneous epochs. Thus, the epoch
E has the form:
74
New Statistical Methods for Dating
Chapter 2
E - T KTH T T K T H T K T T NTH T T PT C
P C P n n C
C P
P
1600 B.C. 753 B.C. 82 B.C. A.D. 250 A.D. 962 A.D. 1619
It is obvious that E contains a repetition, duplicating each other’s epochs.
Besides, it decomposes into the sum of four almost identical copies of shorter
chronicle lines (see Fig. 9). We can schematically write that E = C\ + Ci +
Cz 4- C4. Line C\ is obtained by gluing the chronicles Co and C" together.
Thus, all four chronicle lines Ci, C2, C3, and C4 are practically identical,
being only different in their position on the time axis.
5.10. Dependent dynasties in the Bible and parallel with European
history
There are also other pieces (chronicles) in the GCD containing duplicates.
Consider an example: the chronicle line B embracing the events from 4000
to 586 B.C., described in the Old Testament. We borrowed their chronology
from Blair’s traditional tables (Tables 1-7 in [1]; more precisely, see the data
from Columns 1 and 2 of Tables 1, 2, and 3; Columns 1, 2, and 3 of Table
4; Columns 1 and 2 of Table 5; Column 1 of Table 6; and finally, Column 3
of Table 7). The historical events making up the chronicle line B have been
described in Genesis, Exodus, Leviticus, Numbers, Deuteronomy, the Book of
Joshua, the Book of Judges, the Book of Ruth, the First Book of Samuel, the
Second Book of Samuel, the First Book of the Kings, the Second Book of the
Kings, the First Book of Chronicles, the Second Book of Chronicles, the Book
of Ezra, the Book of Nehemiah, and the Book of Esther. Traditionally, these
events are believed to have occurred in the Near East, i.e., in a region different
from that determined by the events composing the chronicle line E (Europe,
the Mediterranean region; see above). The application of our method for du-
plicate recognition and those methods described in §4 lead to the discovery of
duplicates in J3, which are distributed as follows (for details, see Vol. 2).
B = TKTHTKTKTHTTPTC a ,
n
P
where the epoch C a is part of C. It is not accidental that we have employed
the same symbols in describing the biblical chronicle B as for the European
chronicle E. We see that B coincides with a certain part of E> i.e., overlaps
it (there is parallelism of events). Namely, the following equality is valid:
References
75
H
n
E = TKTNT(TKTHTKT KTPTTPTC)
P C P T n
C
P
or
£ = T K T H T
(chronicle B = Old Testament)
P C P T n
C
P
The length of B equals ca. 2,300 years. Thus, the complete “textbook of
modern history” , i.e., the GCD, contains not only shortened redated chronicles
of the forms E and fl, but also parallel, or isomorphic, i.e., nearly coincident
chronicles of considerable length. Meanwhile, they are traditionally treated
today as chronicles describing different historical epochs.
The following general result is valid. The whole GCD, and not only the
above chronicles E and 2?, can be completely restored from its lesser part Co
describing the events placed to the right of A.D. 300, it being important that
most of the events in Co are placed, in reality, even to the right of A.D. 960.
In particular, the chronicle B can be practically completely restored from its
lesser part which describes the events from A.D. 960 to 1400.
References
[1] Blair, J., Blair’s Chronological and Historical Tables from the Creation to
the Present Time , etc . G. Bell & Sons, London, 1882.
[2] Ginzel, F., Handbuch der mathematischen und technischen Chronologic ,
etc . Leipzig, 1906-1914.
[3] Fomenko, A.T., “On the computation of the second derivative of the moon’s
elongation”, in Controllable Motion Problems: Hierarchical Systems. Perm
University Press, Perm, 1980, pp. 161-166 (in Russian).
76 New Statistical Methods for Dating Chapter 2
§6. A New Empirico-Statistical Procedure for Text Ordering and
Its Applications to the Problems of Dating 1
6.1. The chapter generation
This section presents one of the new methods for dating ancient events worked
out on the basis of statistical principles which I initially formulated and veri-
fied in [1] and which I presented at the Third International Vilnius Conference
on Probability Theory and Mathematical Statistics.
The goal of the method discussed in this section is to find a chronologically
correct order of separate fragments of historical texts and to discover among
them various duplicates, or repetitions, i.e., parts describing the same events.
We call the fragment of a historical text describing the events of (approxi-
mately) one generation a chapter generation (or simply chapter). Let a histor-
ical text X embrace the events in a sufficiently large time interval (A, B), i.e.,
from a year A to a year B. Assume that this text is broken (or can be broken)
into separate chapters X(T), where T denotes the number of a generation
(historical characters) described in a fragment of the text X(T). Meanwhile,
we assume that numbering of the chapters X(T) is determined by their order
in the text X. The obvious question arises: Have these chapters been ordered
chronologically correctly by the author? If, however, the correct (chronologi-
cal) numeration of the chapters has been lost (is unknown or doubtful), how
can it be restored? In other words, how can the events described in the chap-
ters X(T) be ordered chronologically correctly in time?
6.2. The frequency-damping principle
Let a time interval (A,B) described in a text be sufficiently large, i.e., tens
of hundreds of years long. Then, as I discovered while quantitatively process-
ing the information contained in a large set of concrete historical texts, the
following important circumstance should be taken into account. It turns out
that in the overwhelming majority of cases, different historical characters bear
different full names in the text. This can be explained easily, though. As a
matter of fact, a chronicler is interested in distinguishing between different
historical characters in order to avoid any ambiguity. The simplest method to
achieve this is to give different full names to different characters. The fact can
be justified by checking experimentally.
We now formulate the theoretical frequency- damping principle.
In the chronologically correct ordering of chapter generations of a text X,
the author changes historical characters while proceeding from the descrip-
tion of the events of one generation to those of the subsequent one. Namely,
when describing those generations prior to a fixed one numbered To for a
given ordering of chapters, the chronicler mentions no characters of Tq. With
1 First published as an article in DAN SSSR , 268, 6(1983), pp. 1322-1327 (in Russian).
§6
Text Ordering and Problems of Dating
77
a chronologically correct ordering of chapters, this can be explained by the
simple fact that these personages have not yet been born. Then, when de-
scribing To, the chronicler speaks of the historical characters of this generation
most often in chapter X(To). This is quite understandable, for the historical
events described by the author are related to the personages born at that
time. Finally, proceeding with the description of subsequent generations, the
chronicler mentions the characters preceding the generation To less and less,
which is also natural, because the author describes the new historical events
of subsequent centuries whose personages certainly overshadow the deceased
characters and the memories of them.
Since, due to the above remark, we can assume that the “identity” name =
historical character is valid (see above), we shall now investigate the totality
of all full names of personages mentioned in a text under investigation. As a
rule, the term “full” will be omitted.
Consider the set of the names of personages first appearing, for a given
ordering of chapters, in a chapter To of a text X . Denoting the number of
mentions of all the names in the chapter X(Tq) by if (To, To), we count each
name with its multiplicity and calculate the frequency of its being mentioned.
We then see how many times these names have been mentioned in a chapter
X(T) and obtain a certain number K(Tq,T). We stress once again that if a
certain name is encountered several times, then all these references are taken
into account.
Thus, for each number To, we obtain a certain numerical graph
K(To,T), where the argument T is variable. We can now reformulate the
frequency- damping principle as follows.
In numbering the chapters chronologically correctly (i.e., chapters describ-
ing the same events), with duplicates being absent among them, each graph of
K(To,T) must have the following (theoretical) form. The function A(To,T)
vanishes to the right of the point To while reaching its absolute maximum at
the point To itself and decreasing monotonically to the right of To (Fig. 16).
The experimental check has completely confirmed (on the average) this
frequency-damping principle for several dozen historical texts with a pre-
scribed chronologically correct ordering of chapters (see [5]).
6.3. The method of finding the chronologically correct order of
chapters in a historical chronicle
We now describe the method of finding the chronologically correct order of
chapters in a historical text X (or in a whole set of texts). Number all the
chapters of the text X in a certain order, e.g., in which they occur in the
text itself. We then determine the graph of A'(To,T) described above for each
separate chapter X(To). The number of these graphs will equal that of the
chapters in the text X. All these values A(To,T) (for the variables To and T)
are naturally organized into a square matrix K{T} of order n x n, where n is
the total number of chapters in the text.
In the ideal (theoretical) case, the matrix K{T} has the form shown in
78
New Statistical Methods for Dating
Chapter 2
m
Figure 16. The frequency-damping principle (for the case of chronologically
correct ordering of chapters in the chronicle).
Fig. 16a. Namely, all the absolute maxima (lines and columns of the matrix)
are concentrated on the principal diagonal. Then, the farther off the principal
diagonal, the smaller (monotonically) the values K(To,T). The computational
experiment has shown for real historical texts that, with a chronologically
correct ordering of the chapters in a text X, the numbers K(T 0 ,T) decrease,
on the average, monotonically not only with respect to the rows of the matrix
K{T} but also to its columns (see Fig. 16b).
In other words, the frequency of names (personages) of prior origin, i.e.,
from the earlier chapters X(T ) mentioned in the fragment X(To), gradually
decreases as the generation T creating them moves farther away from the
generation To under investigation. It thus turned out that an increase in the
age of a historical character (name) almost always causes a decrease in the
frequency of references to this personage (name) in the subsequent chapters
X(Tq). To estimate the rate and character of the frequency-damping graph
for the references to a name, we can make use of the following averaged graph,
§6
namely,
Text Ordering and Problems of Dating
79
tfav.(t)
J2i-T Q =t K(Tp f i)
where t = 0,1,2, ... , n — 1.
It is clear that it is obtained by averaging the square matrix K{T} with
respect to all diagonals parallel to the principal.
Certainly, the experimental graphs K(To,T) may turn out not to be coin-
cident with the theoretical graph for a concrete text.
It is obvious that, upon varying the original numbering of chapters X(T),
the matrix K{T} and its entries also vary. As a matter of fact, there occurs
a rather complicated redistribution of the names first appearing in a certain
chapter A (To). Let us change the order of chapters of the text X by means of
various permutations, which we denote by <r. We also designate the new chap-
ter numeration corresponding to a permutation <r performed by <tT. While
calculating the new matrix K{<rT} for each of these chapter permutations,
we will seek o*, i.e., an order a of the text chapters, such that all or almost
all frequency graphs of references to the names K(Tq,T) will have the almost
theoretical form shown in Fig. 16. In particular, we will try to make the graph
X av .(0 maximally close to the ideal, monotonically damping graph in Fig. 16.
The order of the textual chapters, for which the deviation of the experi-
mental matrix from the theoretical (damping) is the least, should be taken as
chronologically correct and required.
This method of chapter ordering permits us to date ancient events. In fact,
let a certain historical text Y be given for which it is only known that it
describes some events from a historical epoch (A, B). Assume that we already
have another dated text X describing the same epoch more or less completely.
Let X be separated into the chapter generations X(T). How can we learn
which generation exactly has been described in the text Y in question? We
shall make use of the text X. Add Y to the collection of chapters X(T) of X,
for which it suffices to assume that Y is a new chapter of X, and ascribe a
certain number Tq to it, i.e., insert the chapter Y in place of Tq in the text X.
Then, employing the above method, we find the optimal, i.e., chronologically
correct order of all the chapters of the text X with the chapter Y added.
Meanwhile, we shall therefore also find a chronologically correct place for the
new chapter Y . The relative position which the text Y will occupy among
other chapters of X should evidently be taken as the one desired. We thereby
date the ancient events described in Y relative to the chapters of the text X.
This dating method has been checked against historical texts with an a
priori known dating of the events described. The efficiency of the method has
been fully confirmed (see [5]).
6.4. The frequency- duplicating principle and the method of
duplicate recognition
We now account for a new method of duplicate recognition in a text and
describe the frequency- duplicating principle. Let a historical epoch (A, B) be
80
New Statistical Methods for Dating
Chapter 2
described in some text X which is divided into individual chapters X(T).
Assuming that they have been numbered generally in a chronologically correct
manner, we suppose that there are two duplicates among the chapters, i.e.,
two chapters or fragments of the text, describing the events of one and the
same generation. In other words, these chapters repeat each other, but are
placed by the chronicler in different locations in the text X. Consider the
simplest situation where the same chapter is repeated in X twice, numbered
To as a chapter X(To) and Co as a chapter X(Cq). We take To < Co.
K(C 0 ,T)
• • • ' -•
1 r 0 C 0 n
Figure 17. A new method of duplicate recognition (frequency- duplication prin-
ciple).
It is evident that the frequency graphs I<(To,T) and K(Cq,T) have the
form represented in Fig. 17. The first graph /C(To,T) clearly does not satisfy
the frequency-damping principle (two maxima). Hence, we have to permute
the chapters of the text somehow to achieve better agreement with the the-
oretical damping graph in Fig. 16. Furthermore, we can see that the second
graph vanishes, i.e., /f(Co,T) = 0, which is explained by the fact that there
are no new names appearing in the chapter A (Co) for the first time (they all
have already appeared in the earlier chapter X(To)). It then becomes evident
that the best coincidence of the experimental frequency graph with the theo-
retical one in Fig. 17 is achieved when we juxtapose these two duplicates (i.e.,
chapters X(T 0 ) and X(Co)) or simply identify them.
Thus, if the chapters of the text, which in general are numbered chronolog-
ically correctly, contain two whose frequency graphs have the form approxi-
mately represented in Fig. 17, then they are probably duplicates and should
be identified. This is exactly what we call the frequency- duplicating principle.
A similar reasoning is also valid for the case of several duplicates in a text.
Text Ordering and Problems of Dating
81
§6
In the experiment which I performed, the discovery of such double peaks of
the frequency graph (which correspond to the duplicates) occurred as follows.
Let a,ij be an element of the matrix JFC{T}, placed in the ith row and the
jth column. Consider the matrix {a Q p} consisting of the elements a a £, where
a > i and /? < j, i.e., part of the large matrix K{T } bounded by the ith row
and the jth column. We construct the averaged frequency graph (t) for
it by averaging the values positioned in the matrix {a a /?} on the diagonals
parallel to the principal one. We now assume that the ith and the jth columns
of the frequency matrix K{T} correspond to two duplicates X(i) and X(j)y
i.e., T = i or T = j. Then the averaged frequency graph of K % J V {t) has the
form represented in Fig. 17, i.e., it possesses two maxima.
Then, by marking all those elements a,j (where i < j) in the large matrix
K{T} for which the averaged graph of K*J V (t) has such an anomalous form, we
discover those chapters which may be duplicates. It was required in concrete
computations that the averaged graph of /f]|£(t), where p = i+s and q = j — s,
on the average should be monotonically decreasing if the positive integer s
is sufficiently small compared to the difference j — i . For a more detailed
statistical analysis, see Vol. 2.
6.5. The distribution of old and new duplicates in the Old and New
Testament. A striking example: the Book of Revelation
For many an ancient historical text, commentators have performed the job
of discovering repeating fragments, or duplicates. By a repetition (variant of
a duplicate), we may understand not only a repetition of names (see above)
but, more generally, the repeated description of some event in the text, and
so forth. For example, in the Old and New Testament, all such individual
repetitions (duplicates) discovered by the commentators have been indicated
and collected in the so-called set of parallel passages. This list counts about
20,000 verses in the Old and New Testament. If some historical text X is
supplied with the same (or similar) set indicating duplicates, then, for their
correct chronological ordering, we can apply our method from Section 6.4.
Meanwhile, we have to regard the repeating fragments of the text as repeating
“names” .
The method of text ordering and duplicate recognition is also applicable to
the list of reciprocal citations in any closed collection of historical and other
texts. In particular, I have applied all these methods to the Old and New
Testament, which resulted in the discovery of new and previously unknown
duplicates in addition to those known previously, such as the First and Second
Book of Samuel, the First and Second Book of Kings, and the First and Second
Book of Chronicles. The distribution of these duplicates in the sequence of
the books of the Old and New Testament is represented in Fig. 18 by the line
B y where all the discovered duplicates are denoted by identical letters.
It turned out, in particular, that the traditionally accepted order of chap-
ter generations, and therefore books, of the Old and New Testament differs
sharply from the one chronologically correct in the above sense, which has
82
New Statistical Methods for Dating
Chapter 2
Figure 18. The distribution of duplicates in the Old and New Testament. Com-
parison with the duplicate system in European history.
been discovered by applying our method of text ordering. The averaged graph
of K w .(t) constructed for all the books of the Old and New Testament was
found to be monotonically decreasing, thus satisfying the frequency-damping
§6
Text Ordering and Problems of Dating
83
principle in that and only that case where the separate chapters and books
of the Old Testament are mixed up and permuted in a definite and rather
intricate manner with those of the New Testament. Roughly speaking, the
books of the Old and New Testament can be shifted toward each other.
Meanwhile, it is important that in about 88% of the cases, the mutual
position of chapters and books is retained inside the shifted fragments.
A striking example can be given by the Book of Revelation in the New
Testament, traditionally placed last in the Canon. If this traditional position
were chronologically correct, then its frequency graph K(T,Tq) (constructed
from the frequency matrix column) would have to be of the form represented
by the dotted line in Fig. 19.
18
16
14
12
10
8
6
4
2
Graph of K(T,T 0 ) 16 Experimental graph for
r 0 =218
14
the Book of Revelation
i
>
t.
1 1
1
i 1
lu-
m~ .i'Z
20
40
60 80 100 120
140
160 180 200 218
Figure 19. Strong disagreement between the real frequency graph for the Book
of Revelation ( solid line) and the theoretical frequency graph (dotted line).
However, the real frequency graph for the Book of Revelation (see Fig. 19),
given by a continuous line, is sharply different from the theoretical one. Hence,
to eliminate this discrepancy and to restore the correct chronological order,
we should place the Book of Revelation near the Book of the Prophet Isaiah,
the Book of the Prophet Jeremiah, the Book of the Prophet Ezekiel, Exodus
and Leviticus (with the new ordering discovered by our method).
6.6. Duplicates of epochs in the ‘‘modern textbook” of ancient
history
The same method has also been applied by the author to the written sources of
ancient and medieval Europe, due to which all the duplicate- epochs schemati-
cally represented in Fig. 18 along line E (European history) by identical letters
have been discovered. All these results are consistent with the decomposition
of the GCD (see §5) into the sum of four chronicles, which I discovered on the
basis of quite different statistical dating methods (see §4, §5, and [2]-[5]).
Line E in Fig. 18 schematically represents European history and chronology
(the sequence of events with their traditional dating), while line B shows the
84
New Statistical Methods for Dating
Chapter 2
history described in the Old and New Testament (see §5). The overlapping
(parallelism) of the historical epochs described in the Old and New Testament
and those entered along line E is represented in the listing given below. The
events (epochs) designated in this list by identical letters are, wholly or partly,
duplicates, or fibre into the sum of several duplicates.
Thus, we consecutively describe the basic events making up line E which, as
noted above, in reality includes also the duplicate events from line B marked
by the “=” sign. Thus, this is the table of possible parallelisms, where “=”
means “duplicate” .
(7\) = The Trojan kingdom of seven kings, ca. 1460-1240 B.c.
(T) = The Trojan War, the fall of Troy, ca. 1236-1226 B.C.
(H) = The dynasties of the kings of ancient Greece, ca. 1226-850 B.c.
(T) = The second version for dating the Trojan War (according to the an-
cient authors Hellanic and Damast), one or two generations before the found-
ing of Rome, ca. 850-830 B.c.; landing of the former Trojans in Italy.
= Gn 1-3, Adam, Eve, and Expulsion from Paradise.
(T) = Romulus and Remus, the founding of Rome, the rape of the Sabines,
ca. 760-753 B.c.
= Gn 4:1-16, Cain, Abel.
( K ) = The regal Rome of seven kings (according to Livy), ca. 753-523 B.c.
= Gn 4:17-26; Gn 5:1-31, Enoch, Irad, Methusael, Lamech, Mahalaleel,
and Jared.
(T) = The kings’ exile from Rome, war with the Tarquins, the founding of
republican Rome, 522-509 B.c.
= Gn 5:32, 6, 7, 8, Noah, the Flood, convenants for man to multiply
and to fill the earth, Shem, Ham, and Japheth.
( H/C ) = Ancient republican Rome and ancient Greece, the Persian Wars,
the Peloponnesian War, the Punic Wars, Philip II (king of Macedonia) and the
fall of Byzantium, the empire of Alexander the Great, commander Hannibal,
the end of classical Greece, ca. 509-82 B.C.
= Gn 9, 10, Japheth ’s sons.
(T) = Fall of the republic in ancient Rome, Sulla, Pompey, Julius Caesar,
Augustus Octavianus, civil wars in Italy, 82-23 B.c.
= Gn 11:1-9, building the tower of Babel, dispersing mankind all over
the earth.
( K/P ) = The Roman Empire from 82-27 B.c. to a.d. 217.
= Gn 11:10-32, Arphaxad, Shelah, Serug, Terah, Haran, and Abram.
(T) = Wars and crisis in Italy in the 3rd century A.D., the Gothic War,
Roman “soldier” emperors, anarchy in the empire, Julia Maesa, A.D. 217-251.
= Gn 12, Abram, Sarai, struggle with the Pharaoh.
§6
Text Ordering and Problems of Dating
85
(T) = Restoration of the Roman Empire under Lucius Aurelian, civil wars
in Italy, a.d. 270-306.
= Gn 13, Abram, Haran, separation into two kingdoms.
(K/U/C/P) = The Roman Empire from 306 to 526.
= Gn 14-38, Isaac, Esau, Jacob, Judah, Joseph.
(T) = The Gothic War in Italy in the middle of the 6th century A.D., the
Persian War, fall of the Western Roman Empire, Justinian, Belisarius, Narses,
Totila, a.d. 535-552.
= Gn 39-50, Exodus (rule of Moses), war with the Pharaoh, Leviticus
(legal codification of Justinian), Numbers and Deuteronomy.
(H/n/P) = Medieval papal Rome in A.D. 553-900 and the Carolingians,
the empire of Charlemagne from Pepin of Heristal to Charles the Fat, a.d.
681-887.
= Joshua, his aggressive wars, the Book of Judges 1-18, Israel under
the judges.
(T) = Alberic I and Theodora I, war in Italy, A.D. 901-924.
= The Book of Judges 19-21, the Benjamites and the war with them.
(T) = Alberic II and Theodora II, Italy in A.D. 931-954.
= The Book of Ruth, First Book of Samuel, Second Book of Samuel,
First Book of Kings 1-11, First Book of Chronicles, Second Book of Chronicles
1-9, Saul, Samuel, David, Solomon.
(P/C) = The Holy Roman Empire of the German Nation in Italy and
Germany in 962-1250.
= The First Book of Kings 12-22, the Second Book of Kings 1-23, the
Second Book of Chronicles 10-34, the kingdoms of Israel and Judah.
We now interrupt our listing of the events represented by lines E and B
and stress the agreement of the overlapping of the dynasties indicated in item
(P/C) and the results of another method of dynastic overlapping (see §5).
Namely, according to these results, the dynasty of the ancient kings of Judaea
and that of the medieval rulers of the Holy Roman Empire from 911 to 1307
(note that we take into account the German coronation dates for the Holy Ro-
man emperors) are parallel to each other (overlap) with proximity coefficient
10~ 12 , which indicates their interdependence (see the definition of the prox-
imity coefficient A in §5). Besides, the dynasty of Israel’s ancient kings and the
medieval dynasty composed of the dates of the Roman coronations (and also
of the rule durations) of the rulers of the Holy Roman Empire from 920 to
1170 are also parallel to each other (overlap) with proximity coefficient 10“ 8 .
In both cases, overlapping indicates the dependence of these dynastic pairs
in the sense of §5. In the block of historical events (P/C) which we are now
discussing, the following overlapping occurs. Gregory VII Hildebrand (1020-
1053-1073-1085) is parallel to Jesus (Gk. iesus). This overlapping results from
shifting the dates by 1,052 years. For details, see Vol. 2.
86
New Statistical Methods for Dating
Chapter 2
The chronicle lines E and B then contain the following events.
(T) = The war in the middle of the 13th century in Italy, the fall of the
German dynasty of the Hohenstaufen, establishment of the House of Anjou;
Conrad, Manfred, Charles of Anjou, Conradin, 1252-1268.
= The Second Book of Kings 24-25, the Second Book of Chronicles
35-36, the war with the Pharaoh and Nebuchadnezzar, the fall of Judaea, the
start of Babylonian (Avignon) captivity.
(C) = The Holy Roman Empire of the Habsburgs, 1273-1619, 70-year Avi-
gnon exile of the popes and pontificate from 1305 to 1376, the pontiffs’ return
from France to Italy. The traditional chronology of ancient history is created
at the same time by J. Scaliger (1540-1609) and Dionysius Petavius (1583-
1652). It is possible that Dionysius Exiguus (6th century) is the duplicate,
i.e., the reflection, of Dionysius Petavius (16th century).
(C a ) = a part of (C) and the duplicate is as follows: The Book of Ezra, the
Book of Nehemiah, the Book of Esther (which, taken all together, probably
describe the period from 1305 to 1378), the Babylonian captivity lasts for 70
years, return to Jerusalem.
Thus, the entire GCD (§5) is practically completely restorable from its
lesser parts describing the events only from A.D. 900 to 1650, which is done
by shifting the indicated shorter chronicle backward by ca. 333, 720, 1,053,
and 1,778 years (§5, and Fig. 18 of the present section).
For example, the chronological shift backward by 1,053 years (ca. 1,000
years) could have arisen when the chronologists later compared two different
methods of counting and fixing dates.
(a) By the first method, the dates could have been written as follows, for
example: 13th century A.D. = X.III = Christ 3rd century = christos-III, which
might have indicated the 3rd century since the 11th century A.D., i.e., the
3rd century since the birth of Gregory VII Hildebrand (c/. also the canonical
names of centuries in Italy: trecento, i.e., three hundred, the 14th century A.D.;
quattrocento, i.e., four hundred, the 15th century A.D.) Similarly, the dates
could have also been written thus: A.D. 1500 = 1.500 = Jesus the year 500 =
iesus-500, which might have indicated the 500th year since the beginning of
the 11th century A.D., i.e., since the birth of Gregory VII Hildebrand.
(b) Finally, the second method of counting dates, namely, the years A.D.: It
is possible that the letters X and I were originally supplied by the chroniclers
not with a numerical value (one thousand), but with a meaningful one, i.e.,
they were abbreviations of the names “Christ” and “Jesus” (see above).
In conclusion, we make an addendum to the list of special pairs of dependent
historical dynasties (see §5).
(1) The dynasty of the popes (pontifices) from A.D. 140 to 314 and that of
the Roman popes from A.D. 324 to 532, where A = 8.66 • 10” 8 .
(2) The Holy Roman Empire of the German Nation from A.D. 936 to 1273
and the ancient Roman Empire from 82 B.C. to A.D. 217, where A = 1.3- 10” 12 .
§6
Text Ordering and Problems of Dating
87
References
[1] Fomenko, A.T., “Informative functions and related statistical regularities”,
Abstracts of Communications of the Third International Vilnius Confer-
ence on Probability Theory and Mathematical Statistics. Institute of Math-
ematics and Cybernetics of AN LSSR, Vol. 2, 1981, pp. 211-212 (in Rus-
sian).
[2] Fomenko, A.T., “On the computation of the second derivative of the moon’s
elongation”, in Controllable Motion Problems: Hierarchal Systems. Perm
University Press, Perm, 1980, pp. 161-166 (in Russian).
[3] Fomenko, A.T., “On the properties of the second derivative of the moon’s
elongation and related statistical regularities”, in Problems of Compu-
tational and Applied Mathematics. Vol. 63, AN USSR, Tashkent, 1981,
pp. 136-150 (in Russian).
[4] Fomenko, A.T., “The jump of the second derivative of the moon’s elonga-
tion”, Celestial Mechanics , 25, 1(1981), pp. 33-40.
[5] Fomenko, A.T., “New Experimental and Statistical Methods for Dating
Events of Ancient History and Applications to Global Chronology of the
Ancient World and the Middle Ages”. Pre-print No. B07201 (dated Decem-
ber 9, 1981), Moscow, 1981, pp. 1-100 (English translation is catalogued
in the British Library Department of Printed Books, Cup. 918/87).
CHAPTER 3
New Experimental and Statistical
Methods for Dating Events of Ancient
History, and Their Applications
to the Global Chronology of
Ancient and Medieval History 1
This is a brief review of my book Global Chronology of the Ancient and Me-
dieval World: An Experiment in Statistical Research. Methods and Applica-
tions , containing an account of the results I obtained from 1974 to 1982. The
manuscript is about 6,000 typed pages long, and hence, due to the limited
space, the present chapter is only meant to give the reader an idea of the
essence of the problem, namely, the new methods for dating ancient events
and the construction of the global chronological diagram, a “modern text-
book” for ancient and medieval history, and its decomposition into the sum
of three shifts of four identical chronicles.
I do not pretend to explain completely the problems of purely historical
character. My hypothesis touching upon problems of history may turn out
not to be final but, possibly, require further investigation.
§7. Introduction. N.A. Morozov and Modern Results
The first detailed investigation of today’s problem to substantiate chronology
was carried out by N.A. Morozov, who, between 1924 and 1932, published
the fundamental work Christ [1], in which he gave a detailed criticism of the
traditional chronology of the ancient world. The important fact discovered
in this work is the remaining absence of any substantiation of the accepted
concepts which form the basis for the traditional chronology of ancient times.
Proceeding from the analysis of a large number of facts, Morozov proposed and
1 First published as preprint No. B07201, Moscow, 1981, pp. 1-100 (in Russian).
88
§8
Problems of Historical Chronology
89
partly substantiated his fundamental hypothesis that “traditional” ancient
history had been artificially extended in comparison with “real” events.
Nikolai Aleksandrovich Morozov (1854-1946) is a remarkable universalist
scholar, a well-known Russian scientist, revolutionary, and public figure (for
details, see Nikolai Aleksandrovich Morozov : A Universalist Scholar. Nauka,
Moscow, 1982 (in Russian); V.A. Tvardovskaya, N.A. Morozov in the Russian
Liberation Movement. Nauka, Moscow, 1983 (in Russian); “Nikolai Aleksan-
drovich Morozov” in Bibliography of Scientists of the USSR , Nauka, Moscow,
1981; in Russian). In 1881, he was sentenced to life imprisonment for his po-
litical activity as a member of the revolutionary movement of the Populists.
While in prison, he educated himself in chemistry, physics, astronomy, math-
ematics, and history. After his release, he engaged in scientific activities and
was appointed to a professorship. After the October Revolution, Morozov took
over as director of the Lesgaft Institute of Natural Sciences. This institute was
totally reformed after Morozov left the post of director. In 1922, he was elected
an honorary member of the USSR Academy of Sciences and was awarded the
Order of Lenin and the Order of the Red Banner of Labour. Prom 1924 to
1932, Morozov published his work Christ [1] (the original title was The His-
tory of Human Culture from the Standpoint of the Natural Sciences ), which
stirred up an animated discussion in the press [33], [34], [35]. Although raising
certain serious objections, the critical part of [1] as a whole was not disputed.
In 1974, Prof. M.M. Postnikov read four lectures at Moscow State University,
in which he gave a brief survey of Morozov’s aforementioned work and thereby
drew my attention to the problem of substantiating ancient chronology.
Here, I briefly substantiate the new results I obtained from 1974 to 1982
on the basis of my new methods of dating ancient events. These methods
and certain applications to chronology were published in the scientific papers
listed in the Bibliography.
§8. Problems of Historical Chronology
8.1. Roman chronology as the “spinal column” of European
chronology
First, we shall give a short review of the present state of the chronology of
ancient events. Being an important historical tool, chronology permits us to
determine the time interval between any historical fact and the present if we
translate the units of the chronological data of a document describing this
fact into those of our calendar system, i.e., dates B.c. or a.d.
Many historical conclusions depend on the dates which are events described
in a source under investigation. On changing the dates (e.g., if they are not
uniquely determined), treatment and estimation of the events are also altered.
Present global chronology has been formed as a result of research by several
generations of chronologists of the 16th-19th century. Accordingly, all impor-
tant ancient events were assigned Julian dates. The facts described in any
90 New Experimental and Statistical Methods Chapter 3
recently found old documents are now dated according to a procedure which
we shall illustrate by the following example.
Suppose a Roman consul had been mentioned in some Roman text. Since
the complete list of Roman consuls ranges over a period of more than 1050
years, from Lucius Junius, son of Marcus Brutus, and Tarquinius Superbus
(509 B.c.) to Basil I (a.d. 541) [20], we can adjust the described events to the
time scale by finding the appropriate consul’s name and by referring to the
dates which his consulate has according to the list.
This example is typical in the sense that most modern dating methods
are based on the comparison of data in a document with those whose dating
had been established earlier and is regarded as known and fixed. The above
example was deliberately chosen from Roman chronology. As a matter of fact,
“all the remaining dates in ancient chronology can be related to our calendar
system by means of direct or indirect synchronizations with the Roman ones”
([7], p. 77). In other words, Roman chronology and history is the “spinal
column” of the entire global chronology and history, and it is because of this
fact that in the following we pay special attention to Roman history.
8.2, Scaliger, Petavius, Christian chronographers and secular
chronography
The chronology of ancient and medieval history in its present form was created
and completed to a considerable extent in a series of fundamental works of
the 16th— 19th century beginning with J. Scaliger (1540-1609), the “founder of
modern chronological science” ([7], p. 88; [21]), and D. Petavius (1583-1652)
[22]. We also recommend the chronological works of the 18th— 19th century,
which, though mostly obsolete, contain a lot of useful information [23]-[26].
However, the series of these (and other) works is not entirely complete, since,
as the well-known chronologist E. Bickerman observes, “there is no adequate,
full-scale treatment of ancient chronology” ([7], p. 96, Note 1).
The absence of a modern study which would contain strict scientific sub-
stantiation and the construction of a global chronology of ancient times and
the Middle Ages, based on modern data and methods, may be explained not
only by the enormous historical material still requiring further processing and
revision, but also by objective difficulties mentioned by various authors inves-
tigating at different times the scientific substantiation of chronology.
First, let us point out the fact that the appearance, establishment, and
initial development of chronology occurred within the Church and was com-
pletely under its control for a long time. Ancient chronology is given in the
basic works of J. Scaliger and D. Petavius as a table without any substanti-
ation, church tradition alone being its base. This is not surprising, since “for
many centuries, history has predominantly been a church science, and was
written, as a rule, by the clergy ...” ([8], p. 105).
It is assumed today that the foundations of chronology were laid by Eu-
sebius Pamphili and Jerome (4th century A.D.). The work Eusebii Pamphili
Chronicorum (Chronicle) and that of Jerome were discovered only in the late
§8
Problems of Historical Chronology
91
Middle Ages. Moreover, “the Greek original (of Eusebius’ work — A.F.) now
exists only in fragments and has been completed by the free Latin transla-
tion made by Jerome” (see the Introduction to the Russian translation of the
History of the Church by Eusebius, St. Petersburg, 1848, p. viii). It is curious
that Nicephorus Callistus made an attempt in the 14th century to write a new
history of the first three centuries, i.e., to follow Eusebius’ steps, but “could
do noting more than repeat what Eusebius had already said ([27*], p. xi).
However, the work of Eusebius was published only in 1544 ([27*], p. xiii), i.e.,
later than that of Nicephorus; hence, the question arises whether Eusebius
possibly based his work on Nichephorus Callistus. Beginning with Eusebius
and Jerome, almost everyone of the chronologists until the 16th— 17th century
was often a staunch believer or occupied an official church post (archbishop
Jerome, bishop Theophilus, archbishop J. Ussher, theologian J. Scaliger, etc.).
That is why the numerical data contained in the Bible and a blind following
of church authority originally formed the basis for the chronological approach.
As a result of these cabalistic exercises, there appeared, for example, the fol-
lowing “basic dates” from which the entire chronology of the ancient world
was then constructed: In the opinion of Ussher, the world was created early in
the morning of October 23, 4004 B.c. (see [6]). Furthermore, “secular chronol-
ogy”, which appeared later, was completely based on church dogmas.
“. . . the Christian chronogr&phers put secular chronography into the service of sa-
cred history. . . . Jerome’s compilation became the standard of chronological knowledge
in the West. ...”
“J. Scaliger, the founder of modem chronological science, aimed at reconstruct-
ing the work of Eusebius. . . . The datings of Eusebius, often transmitted incorrectly
(! — A.F.) in manuscripts, are of little use to us today. . .” ([7], pp. 87-88).
Following considerable ambiguity and doubtfulness of these cabalistic cal-
culations, the “date of the creation of the world” varies over a wide range (we
illustrate this with some basic examples): 5508 B.c. (Byzantine date), 5493
B.c. (Alexandrian), 4004 (Ussher, Hebrew date), 5872 B.c. (Septuagint), 4700
B.c. (Samaritan), 3761 B.c. (Jewish), 3941 B.c. (Jerome), 5515 (Theophilus),
5551 (Augustine), and so on ([7], p. 73). The amplitude of these dates spans
2100 years. The problem of the choice of a correct “date of the creation of
the world” is not at all artificial; it is not accidental that it has received
so much attention. As a matter of fact, the overwhelming majority of doc-
uments date the described events by years since the “creation of the world”
without indicating, however, which “creation” they actually mean. There-
fore, the existing multi-millennium discrepancy in the choice of this “date”
considerably affects the dating of all documents of this type. The consecra-
tion of these “dates” by the Church prevented (up to the 18th century) any
critical revision or analysis of the global chronology of ancient times. For ex-
ample, Scaliger called Eusebius’ works “divine” ([27*], p. viii). Being brought
up in the spirit of unquestionable veneration of the predecessors’ authority,
the 16th- and 17th-century chronologists reacted strongly to any “outside”
92 New Experimental and Statistical Methods Chapter 3
criticism. Scaliger himself clearly demonstrated his attitude toward scientific
criticism in the following episode. F. Cimpan writes, 2
“The philologist Joseph de Scaliger, the author of a chronological treatise much
appreciated by the scientific community, has become a passionate quadraturist . . .
(name given to people attempting to construct a square which is equal in area to a
given circle by means of compass and ruler; this problem is, as is generally known,
insoluble— A.F.)” ([36], pp. 149-150).
Scaliger published a book in which he asserted to have established “true
quadrature” .
“However hard the greatest mathematicians of the epoch — Vi&te, Clavius — did try
to prove to him that . . . his reasoning was not correct, everything was useless (it
follows from his ‘proof’ that the perimeter of a regular 196-gon is greater than the
length of the circumcircle containing this 196-gon, which is absurd — A.F.). . . . Scaliger
grouped his admirers, who passionately sustained their opinion, recognized nothing
. . . and addressed them . . . with injuries and epithets full of contempt, affirming that
all geometers had no notion of geometry” ([36], p. 150).
Scaliger first applied (together with Petavius) the astronomical method to
the confirmation (but not at all to the critical verification) of the church
chronology of the previous centuries. The contemporary historians believed
that Scaliger thereby turned church chronology into “scientific” chronology.
This shade of “scientific approach” in combination with church authority
turned out to be sufficient for the 17th- and 18th-century chronologists to
regard the chronological network of available and already considerably con-
servative data as completely reliable. By the 19th century, the chronological
data had expanded so much that it caused respect a priori , at least by its very
existence, so that the chronologists of the 19th century saw as their task only
the introduction of small insignificant changes into the chronology of ancient
times. In the 20th century, the problem is basically regarded as solved, and
chronology has finally been perpetuated in the form moulded by the “scrip-
tures” of Eusebius, Jerome, Theophilus, Augustine, Hippolytus, Clement of
Alexandria, Ussher, Scaliger, and Petavius. For a historian of the 20th cen-
tury, the very thought that chronologists have followed an erroneous scheme
for several hundred years naturally seems to be absurd, for it contradicts
accepted “tradition” and the cultural knowledge assimilated since childhood.
Nevertheless, with the development and liberation of chronology from
church control, serious difficulties arose when new generations of historians at-
tempted to find an agreement between many of the chronological data sources
and the established traditional chronology. Thus, for example, it was discov-
ered that Jerome had made an error of one hundred years ([7], p. 89). The
so-called Sassanid tradition separated Alexander the Great from the Sassanids
by a span of 226 years, and modern historians have increased it to 557 years
(a gap of more than 300 years) ([7], p. 89).
2 Translated from the Rumanian — tr.
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Problems of Historical Chronology
93
As Bickerman notes, “The Jews also allotted only 52 years to the Persian
period of their history, though 205 years separate Cyrus from Alexander”
(according to traditional chronology — A.F.) ([7], p. 89). Egyptian chronology
has also been fragmentarily retained by the Christian chronologists.
The aforementioned list of Manethonian kings has been preserved only in
Christian summaries ([7], p. 82). Some readers probably do not know that
“the Eastern Church avoided the use of the Christian dates, since the date of
Christ’s birth was debated in Constantinople as late as the fourteenth century”
([7], P- 73).
8.3. Questioning the authenticity of Roman tradition.
Hypercriticism and T. Mommsen
Consider Roman chronology based on its leading role in the global chronology
of ancient times. Extensive criticism of “tradition” and its chronology began
as far back as the 18th century at the Academie Royale des Inscriptions et
Medailles founded in 1701 in Paris.
In the 1720s, the general authenticity of Roman historical tradition began
to be questioned (N. Freret, L. de Pouilly). The accumulated material served
as a basis for a more thorough criticism in the 19th century. The well-known
historian T. Mommsen was one of the leading authorities in this branch of
chronology, which was then called “hypercriticism”. He wrote, e.g., that
“King Tarquinius the Second, although he was already grown up at the time of his
father’s death and did not begin to reign till thirty-nine years afterwards, is never-
theless still a young man when he ascends the throne. Pythagoras, who came to Italy
about a generation before the expulsion of the kings (509 B.C. — A.F.), is nevertheless
set down by the Roman historians as a friend of the wise Numa (who died ca. 673
B.C.; the discrepancy amounts to at least 100 years — A.F.). The state envoys sent to
Syracuse in the year 262 transact business with Dionysius the Elder, who ascended
the throne eighty-six years afterwards (348)” ([9], Vol. 3, p. 190).
The traditional chronology of Rome rests on quite a shaky basis. For ex-
ample, between the different versions of dating of such an important event as
the foundation of Rome, there exists a divergence of 500 years ([9], Vol. 3,
p. 190). This oscillation strongly accounts for the datings of a large number of
sources counting years since the “foundation of Rome (City)” (e.g., the His-
tory of Rome by Livy). In general, “Roman traditional history has survived
in the works of quite a few authors; undoubtedly, the most fundamental of
them is Livy’s” ([12], p. 3). It is assumed that Livy was born ca. 59 B.C. and
described 700 years of Rome’s history. Of 142 books, only 35 survive; the first
edition came out only in 1469 and was based on a lost manuscript of unknown
origin; only later a manuscript containing another five books was discovered
in Hessia.
Theodor Mommsen wrote:
u . . . the prospect should be still more lamentable in the field ... of the . . . an-
nals ... of the world The increasing activity of antiquarian research induced the
expectation that the current narrative would be rectified from documents and other
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trustworthy sources; but this hope was not fulfilled. The more and the deeper men
investigated, the more clearly it became apparent what a task it was to write a critical
history of Rome” ([9], Vol. 5, p. 495).
Moreover,
“The falsification of numbers was here (regarding Valerias Antias — A.F.) carried
out down even to contemporary history. ... He (Alexander Polyhistor — A.F.) . . . took
the first steps towards filling up the five hundred years, which were wanting to bring
the destruction of Troy and the origin of Rome into the chronological connection (as a
matter of fact, according to another version of chronology, different from today’s, Troy
had fallen immediately before the foundation of Rome, and not 500 years before it —
A.F.) ... with one of those lists of kings without achievements which are unhappily
common in the Egyptian and Greek chronicles; for, to all appearance, it was he who
launched into the world the kings Aventinus and Tiberinus and the Alban gens of the
Silvii whom the following times accordingly did not neglect to furnish in detail with
name, period of reigning, and, for the sake of greater definitiveness, also a portrait”
([9], pp. 496-497).
A survey of this criticism can be also found in B. Niese [14]. The lengthy
account of the ultrasceptical standpoint questioning the correctness of the
“regal Roman” chronology and, in general, the validity of our knowledge re-
garding the first five centuries (!) of Roman history can be found in [28] and
[29]. (For the difficulties concerning the agreement of Roman documents with
traditional chronology, see [30].)
“As a matter of fact, Roman annals themselves do not survive; therefore, all our
assumptions should be based on the Roman annalists. However, here, too, we face great
difficulties, the main one being that the annalists’ works are also in an unsatisfactory
state” ([12], p. 23).
It is assumed that all ancient Roman dignitaries were recorded year after
year in the Roman fasti . Regarding this, G. Martynov comments:
“But how can the constant controversy about the consuls’ names and, moreover,
their frequent omission or the perfectly random choice of their names be made con-
sistent? ... It is sometimes impossible to understand the fasti, since they are full of
ambiguities. Livy himself was aware of this particular weakness of his basic chrono-
logical principles” ([13], pp. 6-7, 14).
And G. Martynov assumes that
“. . . neither Diodorus nor Livy have correct chronology We cannot have any
confidence in the ‘canvas’ books, from which Licinius Macer and Tubero drew quite
contradictory conclusions. Apparently, upon scrutiny, the most valid documents could
also turn out to have been forged much later ...” ([13], pp. 20, 27-28).
Bickerman states:
“As we have the complete fasti of the Roman consuls for 1 ,050 years ... we can easily
assign Julian years to each of them, provided that the ancient dates are trustworthy
...” ([7], p. 81).
8.4. Difficulties in the establishment of Egyptian chronology
The essential discrepancy found between the chronological data of ancient
sources and the global chronology of ancient times has been discovered in
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Problems of Historical Chronology
95
other ones of its branches too. Thus, considerable difficulties accompanied the
establishment of Egyptian chronology, where a number of documents contra-
dict each other chronologically. For example, while accounting for the his-
tory of Egypt in a consecutive and connected manner, Herodotus places the
Pharaoh Rhampsinitus and Cheops next to each other by calling the former
the latter’s successor. As a modern commentary goes,
“Herodotus mixes up the chronology of Egypt: Rhampsinitus (Ramses II) was
Pharaoh of the 19th dynasty (1345-1200 B.C.), while Cheops was a pharaoh of the
4th dynasty from 2600 to 2480 B.C.” ([31*], p. 513, Comm. 136).
Here, the discrepancy with today’s chronological version attains 1,200 years.
It should be noted that, according to H. Brugsch, Rhampsinitus is Ramses
III, and not Ramses II ([11], p. 325). In general, it turns out that, from the
modern standpoint, the following strange circumstance is valid.
“The chronology of kings due to Herodotus does not correspond to that in the
fragments of the Manethonian list” ([31*], p. 512, Comm. 108).
Usually, Herodotus’ chronology is essentially “shorter”. For example, im-
mediately after a pharaoh of the 4th dynasty, he places another one from the
Ethiopian dynasty, i.e., “he jumps from the end of the 4th dynasty ( ca . 2480
B.c.) to the beginning of the Ethiopian rule in Egypt (ca. 715 B.c.)” ([31*],
p. 516, Comm. 150), closing the 1,800-year “gap”. Note that the choice of a
particular chronological version from a number of mutually contradictory ones
is not always obvious, which, inter alia y illustrates the rivalry between the ad-
herents of the so-called shorter and longer chronology of Egypt, beginning in
the 19th century. The famous Egyptologist H. Brugsch wrote that
“German Egyptologists have attempted to fix the era when Mena, the first Pharaoh,
mounted the throne, with the following results:
B.C.
Boeckh . . . 5702
Unger . . . 5613
Brugsch . . . 4455
B.C.
Lauth . . . 4157
Lepsius . . . 3892
Bunsen . . . 3623
The difference between the extreme dates is enormous, amounting to no less than
2,079 years! ...” ([11], p. 14).
Brugsch wrote further:
“The most authoritative work and researches carried out by competent scientists
in order to check the chronological succession of the Pharaohs and whole dynasties
proved, meanwhile, the inevitable necessity to suggest simultaneous and parallel reigns
in the Manethonian list, which considerably decreases the time required for the own-
ship of the country by thirty Manethonian dynasties. In spite of all the discoveries in
this branch of Egyptology, the numerical data were still in a quite unsatisfactory state
at the end of the 19th century” ([11], p. 14).
The modern tables also assign the coronation of Menes to different dates,
namely to ca. 3100 ([15], pp. 28-29), ca. 3000 B.C. (see the Russian transla-
tion of Bickerman’s Chronology of the Ancient World , Nauka, Moscow, 1975,
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p. 176), and so forth. The total variation is 2,700 years. However, if we take
into account the opinion of the other Egyptologists, the situation becomes
still more serious. Namely, according to Champollion, Menes was coronated
in 5867; according to Le Sueur, in 5770; according to A. Mariette, in 5004;
according to F. Chabas, in 4000; according to E. Meyer, in 3180; according to
T. Andrzejewski, in 2850, according to J . Wilkinson, in 2320; and to E. Palmer,
in 2224 B.c., and so on. The difference between the dating of Champollion
and that of Palmer is 3,643 years. In general, according to Chantepie-de-la
Saussaye, Egyptology first “awoke” only 80 years ago. The results of the re-
search were made public too soon; he spoke of the premature assurance of
Egyptology; chronology was still hesitant at the time [37].
At present, a shorter chronology has been adopted; however, it is extremely
self-contradictory, with the uncertainties still unresolved today.
Still more complicated is the situation with the list of kings, compiled by
the Sumerian priests,
M . . . a sort of skeleton of history ... of our English schoolbooks, but unfortunately
it did not seem to help The entire chronology is palpably absurd” ([32], p. 14).
Moreover, it turns out that the dynastic sequence was set arbitrarily ([32*],
p. 107). The very ancient age ascribed today to these lists contradicts the
archaeological data. In his description of the excavations of regal tombs in
Mesopotamia, L. Woolley speaks of a series of golden toilet sets which were
found, one of the best experts declaring that the things were made by the
Arabs in the 13th century A.D.! And he cannot be censured for the blunder,
adds L. Woolley condescendingly: No one could suppose the craft to be so
skillful in the 3rd millennium B.c. ([32] and [32*], p. 61).
The well-known historian of science Otto Neugebauer wrote:
“With the use of an enormous learned apparatus, the author (Jeremiah — A.F.)
develops the panbabylonistic doctrine which flourished in Germany between 1900-
1914. . . . This school was built on wild theories about the great age of Babylonian
astronomy A supreme disregard created a fantastic picture which exercised (and
still exercises) a great influence on the literature concerning Babylonia” ([40], p. 138).
8.5. Competing chronological versions. De A r cilia, J. Hardouin,
I. Newton and R. Baldauf
It turns out that simultaneously with Scaliger’s and Petavius’ versions of the
global chronology still accepted today, another, competing hypothesis negat-
ing the first, and indicating the “youth” of the “written history” known to
us, was born as early as the 16th century A.D. We illustrate this with the
following citation from Morozov’s book [1]:
“Professor de Arcilla of Salamanca University published two of his works Pro-
gramma Historiae Universalis and Divinae Florae Historicae in the 16th century,
where he stated that the whole of ancient history had been forged in the Middle
Ages; the same conclusions were reached by the Jesuit historian and archaeologist
J. Hardouin (1646-1729), who regarded the classical literature as written by the
monks of the preceding, 16th century A.D. (see his books Consiliorum Collectio regia
§8
Problems of Historical Chronology
97
maxima, Chronologiae ex nvmmis antiqvis restitvtae prolvsio it nvmmis Htrodiadvm,
Prolegomena ad censuram veterum scripiorum). The German Privatdozent Robert
Baldauf wrote his book Historic und Kritik in 1902-1903, where he asserted on the
basis of purely philological argument that not only ancient, but even medieval history
was a falsification of the Renaissance and subsequent centuries” ((l], Vol. 7, p. vii-viii,
Introduction).
The famous Isaac Newton in his Abreges de la Chronologic , containing the
work of forty years of investigation, sharply criticized traditional chronology.
Many events of Greek history were shifted by him forwards by 200-300 years,
which coincides with the first chronological shift I discovered, and is described
in my publications (see above). Many events of Egyptian history were “shifted
forwards” by I. Newton by about 1,800 years, which again, quite suddenly,
coincides with the third chronological shift of the author (see above). The
enormous work performed by Newton was taken skeptically by his contempo-
raries.
W. Whiston wrote:
“Sir Isaac Newton composed a Chronology, and wrote 18 copies of its first and
principal chapter with his own hand . . . which proved no better than a sagacious
romance” ([89], p. 35).
8.6. Tacitus and Bracciolioni. Cicero and Barzizza
It is generally known that most of global chronology was initially constructed
by analyzing the chronological references of ancient written sources; the ques-
tion of their origin then becomes quite interesting. Unfortunately, modern
historiography supplies no complete survey of the circumstances in which
the ancient manuscripts appeared and only notes the general fact that the
overwhelming majority of the documents were discovered as early as the Re-
naissance, years after the “dark ages”. It is clear that the manuscripts often
appeared in an environment which was not conducive to speeding up a critical
analysis of the datings.
The two well-known 19th-century historians P. Hochart from France and
J. Ross from England “proved” in 1882-1885 and 1878, respectively, that the
History of Tacitus Cornelius had been actually written by the well-known
humanist Poggio Bracciolini [38, 39]. Without discussing the authenticity of
the History (in our opinion, it is authentic, has not been falsified, and de-
scribes authentic events), we note that the history of its discovery really led
to considerable controversy. It was Poggio Bracciolioni who had discovered and
published the works of Quintilian, Valerius Flaccus, Marcellus, Probus, certain
treatises of Cicero, Lucretius, Petronius, Plautus, Tertullian, St. Marcellinus,
Calpurnius Siculus, and so forth [38, 39]. Besides the original manuscripts,
Bracciolioni traded in copies which he sold for enormous sums of money. For
example, after having sold a copy of Livy, Bracciolioni bought himself a villa
in Florence. The circumstances of these findings and the manuscripts’ dating
have never been made clear.
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In the 15th century, Italy saw the arrival of the famous humanists Manuel
Chrysoloras, Georgius Gemistos Pletho, Bessarion of Nicae, and others, who
were first (!) to acquaint Europe with the achievements of the allegedly “an-
cient” Greek thought. The Byzantine Empire then gave the West almost all
the preserved Greek manuscripts of antiquity. In Neugebauer’s words:
“The majority of manuscripts on which our knowledge of Greek science is based
are Byzantine codices, written between 500 and 1500 years after the lifetime of their
authors” ([40], p. 57).
According to traditional history, the whole of classical ancient literature
became known only during the years of the Renaissance or immediately before
this epoch [41]. Analysis shows that the vagueness of its origin and the absence
of documented data regarding it in the preceding, so-called Dark Ages makes
us suggest that the texts had been previously nonexistent [1].
For example, the oldest manuscripts of the so-called incomplete recension
of Cicero’s texts are those of the 9th and 10th century; however, the archetype
of the incomplete recension had been “lost long ago” [42*]. In the 14th and
15th century, Cicero’s works attracted more and more attention, and
44 . . . it came to the Milan Professor Gasparine Barzizza . . . having undertaken in
ca. 1420 A .D. the risky business of filling by himself gaps in the incomplete recension*
to make it consistent (! — A.F.). No sooner had he finished the job than a miracle
occurred: in the sleepy Italian town of Lodi, a lost manuscript with the complete text
of all of Cicero’s rhetorical works was found. . . . Barzizza and his disciples rushed onto
the new find, deciphered its ancient script (probably of the 13th century A.D.), and,
finally, made a copy which could be read easily. The transcripts were made from this
one, and they are the ones which make up the Complete recension 1 . . .. Meanwhile,
the incorrigible took place: the archetype of the recension, i.e., the Lodi manuscript,
turned out to be neglected; nobody wanted to rack his brains over its difficult text;
it was sent back to Lodi and lost there forever; since A.D. 1428, nothing has been
known about its fate. The European philologists deplore the loss even today” ([42*“],
pp. 387-388).
Suetonius’ De vitis XII Caesarum is also only available in much later tran-
scripts, all originating from “one antique manuscript” allegedly possessed by
Einhard, who, while creating his Vita Karoli Magni (Life of Charlemagne) in
ca. A.D. 818, carefully reproduced the assumingly biographical stories of Sue-
tonius [43]. The Codex Fuldensis and the first copies have not survived [43].
The oldest manuscript of Suetonius is that of the 9th century, but it became
known only in the 16th century, whereas the others are dated to not earlier
than the 11th century A.D.
8.7. Vitruvius and L. Alberti
Ancient sources were dated in the 14th— 16th century on some unknown basis.
The architect Vitruvius’ De architectura was discovered only in 1497. In its
astronomical part, he very precisely listed the sidereal (!) periods of the ma-
jor planets ([1], Vol. 4, p. 624). Allegedly living in the lst-2nd century A.D.,
Vitruvius knew them better than Copernicus. The period of Saturn differs
from the contemporary value by only 0.0007; for Mars, the error is only 0.006,
§8
Problems of Historical Chronology
99
and 0.003 for Jupiter ([1], Vol. 4, pp. 625-626). Note the far-reaching parallels
between Vitruvius* books and those of the remarkable 15th-century human-
ist L. Alberti [44]. (By the way, due to frequently assimilated “v” and “b”,
it may be even conjectured that the names themselves are close: Alb(v)erti-
Vitruvius.) Like Vitruvius, Alberti (1404-1472) was famous as the greatest
Italian architect, author of the well-known architectural theory which is re-
lated, in the closest manner, to the similar theory of Vitruvius [44]. Just like
Vitruvius, he created a fundamental work including not only his theory of
architecture, but also some information from mathematics, optics, and me-
chanics. It is strange that the title of Alberti’s book Di re aedificatoria is
similar to that by Vitruvius. It is now assumed that Alberti modelled his own
treatise on Vitruvius* work [44]. The work of Alberti is wholly in “antique
tones’*. Specialists have long ago constructed tables in which Alberti’s and
Vitruvius’ works are juxtaposed (sometimes coinciding verbatiml).
Thus, the book of Vitruvius (as well as that of Alberti) absolutely naturally
corresponds to the atmosphere and ideology of the 15th century. The over-
whelming majority of Alberti’s buildings were designed “in antique style”. He
erected a palace “after a Roman amphitheatre” . Thus, the leading architect
of the Renaissance fills the cities of Italy with antique buildings which are
regarded now (but not at all in the 15th century) as “imitations of antiquity” .
Alberti wrote books in “antique manner” , not giving any thought to that they
could later be called “imitations of antiquity” by the historians. And it was
only in 1497 that the book of the alleged “ancient” architect Vitruvius was
discovered, which almost verbatim coincides with an earlier and similar book
of Alberti. It is possible that the architects of the 14th- 15th century did not
regard their activities as an “imitation of antiquity”, but created it. However,
the theory of “imitation” will appear in the works of much later researchers.
It is probably due to the chronological shift of some of the medieval docu-
ments, buildings, etc., into oldest antiquity. In particular, this could have led
to “doubling” Alberti, trebling Pletho, and so forth.
8.8. “The chaos of medieval datings” (E. Bickerman). Medieval
anachronisms and medieval concepts of time
The history of finding and publishing ancient scientific treatises in all its
essential features is similar to that of manuscripts in the humanities. M. Ya.
Vygodsky wrote:
“No antique manuscript of the Elements by Euclid has survived. . . . The most
ancient of the manuscripts known to us is a copy made in A.D. 888. . . . There are
many manuscripts dated by the 10th-13th-century” ([87], p. 224).
I.G. Bashmakova informs us that, even before the first Latin translation of
Arithmetica,
“. . . the European scientists had made use of Diophantus’ algebraic methods, not
being familiar with his works” ([111], p. 25).
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Bashmakova characterizes this situation just as “somewhat paradoxical”.
And I.N. Veselovsky says that the basis for all modern editions of Archimedes’
works is a lost manuscript of the 15th century and the so-called Constantinople
palimpsest found only in 1907. It is assumed that Archimedes’ manuscripts
first appeared in Europe after 1204 ([112], pp. 54-56).
“The Conics of Apollonius were published only in 1537, his works having been
published only after the death of Johann Kepler, who had discovered the importance
of the described objects” ([94], p. 54).
The chronological version of Scaliger was not at all unique. In general,
Bickerman deploringly speaks of “the chaos of medieval datings” ([7], p. 78).
Besides, the analysis of ancient documents demonstrates that the earlier ideas
of time were greatly different from the modern. Before the invention of the
clock, time had been regarded as “anthropomorphic”, and the character of its
course depended (!) on the character of the events.
Until the 13th— 14th century, devices for measuring time were a rarity and
luxury [45].
“The sundial . . . sandpiece and clepsydra were common in medieval Europe, with
the sundial being suitable only when it is fine, and the clepsydra remaining a rarity”
([8], p. 94).
At the end of the 9th century A.D., candles were widely used to measure
time; for example, the king of England, Alfred, took candles of equal length
when travelling, and ordered to light them one after another ([8], p. 95). The
same time count was employed even in the 13th and 14th century, for example,
at the time of Charles V.
“The monks orientated themselves by the pages of the Sacred books or psalms read
between two observations of the sky” ([8], p. 94).
For exact astronomical observations, a clock with the second hand was
required, but
“. . . even after the invention and wide use in Europe of the mechanical clock, it did
not have the minute hand for a very long time” ([8], p. 95).
Otto Spengler in his The Decline of the West asserted that the mechanical
clock had been invented ca. A.D. 1000 by A. Gerbert. However, A.Ya. Gurevich
asserts that Gerbert only made the clepsydra more sophisticated, and that
“. . . the mechanical clock was invented at the end of the 13th century A.D.” ([8],
pp. 134-135).
It is generally known that a precise time piece is required for astronomical
observations. However, the following strange phenomenon occurred: Appear-
ing in China, the Europeans acquainted the Chinese with some of their own
inventions. The mechanical clock did interest the Chinese rulers, but not as
a precise time piece — just as a funny toy! ([88], pp. 80-87). And it is tradi-
tionally believed that precise Chinese astronomy had blossomed thousands of
years B.C.!
§8
Problems of Historical Chronology
101
The most intricate chronological cabbala was born and developed in the
Middle Ages in a paradoxical contrast with the inaccuracy of measuring real
time. In particular,
“. . . the same time intervals which are used for measuring earth . . . time acquire
quite a different duration. . . . When used to measure biblical events . . , Augustine
equalized each day of the Creation to a millennium (! — A.F.) and attempted to deter-
mine the duration of the history of mankind. . . ” ([8], pp. 109-110).
Such an intrinsic feature of medieval historiography as anachronism is very
important for us.
“The past is represented in terms of the same categories as in modem times. . . .
The biblical and ancient historical figures are dressed in medieval costumes. . . . De-
picting kings and patriarchs of the Old Testament with ancient sages and evangelical
personages side by side on cathedral portals discloses the anachronistic attitude to-
ward history best of all. . . . The crusaders were convinced at the end of the 11th
century that they castigated not the descendants of the Saviour’s murderers, but the
murderers themselves. . .” ([8], pp. 117-118).
Proceeding from traditional chronology, the modern historians believe that
the Middle Ages “mixed up epochs and notions” on a very large scale, that
the medieval authors identified ancient biblical epoch with that of the Middle
Ages only because of their “ignorance”. But, besides the traditional explana-
tion (alleged inexplicable “love for anachronisms”), another point of view is
possible, namely, that all these statements of medieval authors, which seem
strange now, are authentic and are regarded as “anachronisms” only because
we follow another chronology.
Traditional chronology (Scaliger’s version) only fixed one of several me-
dieval chronological conceptions. Along with the chronology adopted today,
other versions existed earlier, too. For example, the 10th-13th-century Holy
Roman Empire was regarded to have immediately followed the Roman Em-
pire, which, according to the modern ideas, occurred in the 6th century A.D.
([46], Vol. 1, p. 16). Here are the traces of the strange medieval controversy
(from the modern standpoint). Assumingly basing himself on a whole series of
philological and psychological observations, Petrarch asserted that the privi-
leges given by Caesar and Nero to the Austrian duchy (in the 13th century
a.d.! — A.F.) were forged. (Then it had to be proved.) ([46], Vol. 1, p. 32).
For a modern specialist, the very thought that Caesar and Nero meant the
Austrian duchy who began ruling only in a.d. 1273, i.e., 1200 years later, is ab-
surd (ibid.). But Petrarch’s 14th century opponents did not think so (“it had
to be proved at that time” [46]. Regarding the same documents, E. Priester
noted that all the interested people understood quite well that they had obvi-
ously been falsified and shamelessly forged (this being today’s point of view);
nevertheless, they “politely” turned a blind eye to this circumstance [17].
For example, the reader is accustomed to the thought that the famous
gladiator fights took place only in “the far-away past”. But this is not so.
After writing about the gladiators’ fights in ancient Rome, V.I. Klassovsky
adds immediately that they also occurred in 14th-century Europe [47]! He
102 New Experimental and Statistical Methods Chapter 3
also points to fights in Naples in ca. A.D. 1344 ([47], p. 212). As in antiquity,
these medieval fights ended in the death of a fighter [47].
8.9. The chronology of the biblical manuscripts. L. Tischendorf
Dating of religious sources remains very much unclear. The chronology and
dating of the biblical books are quite uncertain and are generally known to be
based on the authority of the Christian theologians of the late Middle Ages.
“The most ancient of more or less complete surviving copies of the Bible are the
Code x Alexandrinus, Codex Vaticanus , Codex Sinaiticus. ... All three manuscripts
. . . are dated (paleographically, i.e., on the basis of comparing the handwriting style*
of the manuscript with that of other manuscripts whose dating is taken as known a
•priori — A.F.) ... by the second half of the 4th century A.D. The language of the
Codices is Greek. . . . The lesser known is the Codex Vaticanus ; in particular, it is not
clear how and whence this source appeared in the Vatican in ca. 1475. ... It is known
about the Codex Alexandrinus that . . . the patriarch Cyrilles Lucaris presented it as
a gift to the king of Great Britain and Ireland, Charles I, in 1628” ([48], pp. 267-268).
The Codex Sinaiticus was discovered only in the 19th century by L. Tis-
chendorf ([48], pp. 268-270). Of the same type of sources are the Codex Bezae
and Codex Ephraemi Syri Rescriptus. These are palimpsests, i.e., writing ma-
terials which were cleared and reused. A scribe erased the biblical text from a
parchment allegedly somewhere between the 12th and the 13th century, and
wrote instead the work of Ephraem Syrus (note how expensive the parchment
was!).
It is due to this document that Tischendorf made a name for himself by
“dating” it to the 5th century A.D.
Thus, all three of these most ancient codexes were actually discovered only
after the 15th century. The reputation of these documents as being ancient was
based on Tischendorf^ authority, judging by the “handwriting style” . How-
ever, the idea itself of paleographic dating assumes that the global chronology
of other documents is known a priori ; hence, this dating method is not inde-
pendent.
Of the individual biblical works, the most ancient one is the manuscript of
the Book of the Prophet Zechariah and that of Malachi, dated to the 15th
century A.D. (also paleographically) [48].
We should also bear in mind the strange fact that “the most ancient of the
surviving manuscripts of the Bible have been written in Greek” ([48], p. 270).
No Jewish manuscripts of the Bible dated to earlier than the 9th century
A.D. (!) exist, though those of later times, mainly of the middle of the 13th
century A.D., are kept in many national libraries. The most ancient Jewish
manuscript containing the complete text of the Old Testament is referred only
to a.d. 1008 ([48], p. 270).
The biblical Canon is assumed to have been established by the synod at
Laodicea in a.d. 363; however, no acts of this and earlier synods survived [49].
In reality, the Canon was officially established only since the new Council of
Trent had assembled in 1545 and lasted, with intervals, until 1563 (during
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Problems of Historical Chronology
103
the Reformation). By the order of the Council, many books were regarded as
apocryphal, in particular, the Book of the Kings of Judah and Israel ([50],
p. 76).
It turns out that most datings of biblical manuscripts are overwhelmingly
based on paleography. We have pointed out that this depends on a global
chronology assumed to be known a priori. We illustrate this with the following
example.
“In 1902, the Englishman Nash bought in Egypt a fragment of a Jewish manuscript
written on a papyrus, the dating of which the scientists cannot agree upon until today”
([48], p. 273).
Finally, it was agreed that the text referred to the turn of our millennium.
M . . . after the Qumran manuscripts had been discovered, the comparison of the
handwriting’ on the Nash papyrus and the Qumran scrolls made it possible to estab-
lish at once that the latter were quite old” ([48], pp. 272-273).
Thus, one scrap of the papyrus, whose dating was not “agreed upon by
everyone”, influenced the dating of all the other documents. Nevertheless,
. . dating the manuscripts (of Qumran — A.F.) led to much controversy among
the scientists (namely, from the 2nd century B.C. to the time of the crusaders)” ([50],
p. 47).
For example, the American historian S. Zeitlin categorically insists on the
medieval origin of the texts ([48], p. 27). It is assumed today that the original
dating of the Qumran scrolls to the turn of the millennium was confirmed by
radiocarbon dating. However, as will be shown below, it is extremely unstable
for comparatively small time intervals, and the application to events which
are 2-3 millennia old is quite problematic due to the obtained “scattering” by
up to about one thousand years (for the indicated time interval).
8.10. Vowels in ancient manuscripts
In the attempt to read and date most of the ancient, medieval, biblical, an-
cient Egyptian, and other manuscripts, certain basic problems are frequently
encountered.
As soon as J . Sunderland started investigating the original language of the
Old Testament, he, in his words,
“. . . faced the fact of enormous and even startling importance. The thing is that
the Jewish written language originally had neither vowels nor signs replacing them.
The books of the Old Testament were written only with consonants” ([49], p. 155).
This is also typical for other languages. For example, an ancient Slavonic
text was a chain of only consonants, too; sometimes even without signs re-
placing the vowels, or without division into words. Old Egyptian texts were
also written in consonants only. According to E. Bickerman,
“. . . the names of Egyptian kings are given in contemporary literature schemat-
ically, in a quite arbitrary, so-called scholastic manner adopted in school textbooks.
These forms are often greatly different from each other; it is impossible to order them
somehow, due to their arbitrary reading (!) which became traditional” ([7*], p. 176).
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Probably, the rarity and high cost of writing materials in ancient times made
the scribes save them, and omit the vowels, thereby essentially shortening the
text. It is also possible that writing out all the words in a row, without intervals
between them, pursued the same goal.
However, if we take the Jewish Bible or a manuscript today, we shall find
in them the skeleton of vowels filled with dots and other signs denoting the
missing vowels. These signs did not belong to the old Jewish Bible. The books
were read by consonants, and the intervals were filled with vowels according
to one’s skill and the apparent requirements of the context and oral legends
[49]. Imagine how exact the meaning of a word written in consonants can be
if, for example, CLN can mean clean, clan, colon, and so forth!
According to T. Curtis, even for the priests, the content of manuscripts
remained extremely doubtful and could be understood only by means of the
authority of the legend ([49], p. 155). It is assumed that this serious short-
coming of the Jewish Bible had been eliminated not earlier than the 7th or
8th century A.D., when the Massoretes revised the Bible and added signs re-
placing the vowels; but they had no manuals, except their own reason, and
a very imperfect legendary tradition ([49], pp. 156-157). S. Driver adds that,
since the times of the Massoretes in the 7th-8th century A.D., the Jews have
taken to keeping their sacred books with extraordinary care, but then it was
too late to repair the damage already done. The result of such attentiveness
was just the immortalization of the distortions, which were then placed on
exactly the same level of authority with the original text ([49], p. 157).
The opinion reigning earlier was that the vowels had been introduced into
the Jewish text by Ezra in the 5th century B.C. But in the 16th and 17th
century, E. Levita and J. Capellus in France refuted this opinion and proved
that the vowels had been introduced only by the Massoretes. The discovery
created a sensation in the whole of Protestant Europe. Many people believed
that the new theory would lead to disproving the religion completely. If the
vowels were not a matter of Divine Revelation, but only a human invention,
besides, a much later one, then how could we rely on the text of the Scripture?
This discussion was one of the hottest in the history of the new biblical criti-
cism and proceeded for more than a century, stopping only when the validity
of the new point of view was acknowledged by everyone ([49], pp. 157-158).
8.11. Traditional biblical geography
Even if the vowels of common words are not that important, the situation
changes completely when their combination meaning a city, country, the name
of a king, etc., appears in an ancient text. Tens and hundreds of different vari-
ants of vowels for one term may be found, starting the “identifications” of the
biblical vowel-free names of cities, countries, and others, made by traditional
historians proceeding from the chronological version of J . Scaliger and the lo-
calization referring the biblical events to the Near East. As the archaeologist
M. Burrows notes, the archaeological job generally leads to the undoubtedly
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Problems of Historical Chronology
105
strongest creed in the reliability of biblical information [48]. F. Kenyon of
the British Museum insists as much categorically on archaeology refuting the
“destructive skepticism of the second half of the 19th century”. (W. Kelley
has published a book with the remarkable title Und die Bibel hat dock Recht ,
asserting that the Bible was right anyway.)
But here is information reported by the well-known archaeologist G. Wright,
who, by the way, is a staunch partisan of the correctness of orthodox local-
ization and of dating biblical events. He wrote,
“A great many findings do not prove or disprove anything; they fill the background
and only serve as historical artifacts. Unfortunately, the desire *to prove’ the Bible
permeates many works available to the average reader. Historical evidence may be
used in an incorrect manner, whereas the conclusions drawn are often erroneous and
only half correct” ([48], p. 17).
If we attentively examine the concrete facts, then we shall see that none
of the books of the Old Testament contain any solid archaeological confir-
mation of their traditional geographical and time localization. The whole
“Mesopotamian” biblical theory will be questioned [1],
I.A. Kryvelev wrote,
“The traditional localization of the events described in the New Testament is no
better. The reader interested in biblical archeology may be bewildered by the hun-
dreds of pages speaking of excavations, landscapes, or artifacts, historical and bibli-
cal background. And, in conclusion, when it comes to the results of the whole job,
there are only a number of indistinct and imprecise statements about the problem
not having been completely solved, but that there is still hope for the future, and
so forth. We may be absolutely sure that none of the stories of the New Testament
contains any somewhat convincing archaeological confirmation (in terms of the tradi-
tional localization — A.F.). This is perfectly true, in particular, if applied to the figure
and biography of Jesus Christ. Not a single spot traditionally regarded as the arena of
a particular event occurring in the New Testament can be indicated with the slightest
degree of confidence” ([48], pp. 200-201).
We note at the same time that, along with the traditional point of view
which as we have seen has no sure confirmation, there also exist other, compet-
ing versions. For example, beginning with the 13th century A.D., the Catholic
Church declared that, in the Italian town of Loreto, there was the very house
where the Virgin Mary lived and where she met the archangel Gabriel ([48],
p. 198). It turns out that “Loreto is a place of pilgrimage of Catholics up to
the present day” ([90], p. 37).
8.12. Problems of geographical localization of ancient events
Considerable difficulties accompany the attempts of geographical localization
of many of the ancient events and cities. For example, Naples (which means
simply “new town”) figured in many ancient chronicles in several images, and
in essentially different geographical regions, namely, Naples in Italy, Carthage
(which also means “new city”) [51], Naples in Palestine [52], Scythian Naples,
and others. One of the localizations of the famous city of Troy is near the
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Hellespont (for which, though, there are also several quite different geograph-
ical localizations). It is for this particular reason that Schliemann ascribed
the name of Troy to the city he excavated near the Hellespont. According to
traditional chronology, Troy was completely destroyed in the 12-1 3th century
B.c. [7]. But, in the Middle Ages, Italian Troy, which still exists today [53],
enjoyed widespread fame. This is the celebrated medieval city which played
an important role in many medieval wars; especially, in the well-known war
of the 13th century. Many Byzantine historians also speak of Homer’s Troy
as of an existing medieval city, namely, Choniates Nicetas [54] and Gregoras
Nicephoras [55].
Livy indicates the spot named Troy and the TVojan region in Italy ([56],
Vol. 1, pp. 3-4, Bk. 1). Certain medieval historians identified Troy with Jerusa-
lem ([57], pp. 88, 235, 162, 207), which embarrasses the modern commentators:
“The book of Homer somewhat suddenly turned (in the medieval text, while de-
scribing Alexander’s expedition to Troy — A.F.) . .. into the book on the destruction
of Jerusalem” ([57], p. 162).
Anna Comnena, speaking of Ithaca (homeland of Homer’s Odyssey ), de-
clares that a large city called Jerusalem has been built on the island of Ithaca
([58], Vol. 2). The second name of Troy is Ilion, whereas the second name
of Jerusalem is Aelia Capitolina ([1], Vol. 7). Thus, in the names of these
cities, there is a similarity: Aelia — Ilion. Eusebius Pamphili told that two
small Phrygian towns with totally different names were called Jerusalem ([1],
Vol. 7, p. 893)! It is traditionally assumed that Great Greece always was on
the Balkan Peninsula; however, in the Middle Ages, the south of Italy popu-
lated by Greek colonies was also called Great Greece [27]. It is also believed
that the town of Babel was situated in modern Mesopotamia. Certain ancient
texts are of a different opinion. For example, the medieval Serbian text of the
Romance of Alexander the Great places Babel in Egypt; moreover, accord-
ing to this text, Alexander the Great died in Egypt, whereas he is believed
to have deceased in Mesopotamia according to the traditional version ([57],
p. 255). Furthermore, Babel is the Greek name of a city located opposite the
pyramids. During the Middle Ages, Cairo was sometimes called by this name.
Nowadays, it is a suburb of Cairo [59]. The term “babel” has a meaning-
ful translation (as have the names of many other cities); hence, it has been
applied to different towns by different chroniclers. That Rome was called Ba-
bel in the Middle Ages was reported by Eusebius [27]. Moreover, by Babel,
the Byzantine historians (in the Middle Ages!) more often than not meant
Baghdad [60]. The medieval Byzantine author of the 11th century, Michael
Psellus, speaks of Babel as of an existing and not at all destroyed city [60].
In many documents, there is a confusion between two Romes, namely, Rome
in Italy and New Rome (i.e., Constantinople on the Bosphorus). Both Romes
are the capitals of the two famous empires, the Western and Eastern ones.
The citizens of New Rome stubbornly called themselves “Romans”. A large
percentage of Byzantine coins were supplied with Latin, and not Greek, in-
scriptions. The capital of the new empire was transferred both from Rome to
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Problems of Historical Chronology
107
New Rome (by Constantine I, ca. A.D. 330) and, vice versa, from New Rome to
Rome (attempt to transfer it in A.D. 663 by Constantine III) [1]. It is possible
that the identification of Constantinople with Troy and the famous “Trojan
war” is a partial reflection of the battle in A.D. 1204 between the crusaders
and the Byzantine Empire (the fall of Constantinople in A.D. 1204).
The complete manuscript of Histories by Herodotus was first discovered
only in the 15th century [31]. This famous manuscript was brought by the
Byzantine scientists to Western Europe after the capturing of Constantinople
by the Turks in 1453. The importance of Herodotus for traditional history
cannot be underestimated. But he declares, quite unexpectedly, that the Nile
flows parallel to the Ister, which is now identified with the Danube (and not
the Dniester). It turns out that “the opinion that the Danube and the Nile are
parallel reigned in Medieval Europe even up to the end of the 13th century”.
Thus Herodotus makes the typical error that is characteristic of medieval
authors.
The identification of Herodotus’ geographical data with the modern map
entails great difficulties within the framework of the events described in tra-
ditional localization.
In particular, numerous corrections which the modern commentators are
forced to introduce if they perform such identification show that “Herodotus’
map” is, possibly, reversed with respect to the modern one (with East and
West interchanged). Such orientation is typical for many a medieval map. The
commentators are forced to assume on different pages of Herodotus’ Histories
the same names for quite different seas. For example, according to the modern
commentators, to retain the traditional localization of events described by
Herodotus, the following “identifications” should be made, namely, Red Sea
= Southern Sea = Black Sea = Mediterranean Sea = Arabian Gulf = Our
Sea = Indian Ocean.
The fact that Herodotus mentions Crestonia and Creston seems to be quite
strange. For Herodotus, there exist even the whole region of Crestonia, the
town of Creston, the country of Crossae, and the district of Creston aei. Ac-
cording to Herodotus, the Crestonaei were invaders from other countries [31].
It is possible that all these references were related to the medieval Cru-
saders (“cross” is also a Crusader term used in the Middle Ages) flooding
Greece in the 12th-13th centuries. (See also works asserting that 15.44 of
Tacitus’ Annals originally had christianos for chrestianos , which also assimi-
lates Herodotus’ terminology to the medieval one [38].
8.13. Modern analysis of biblical geography
Many strange phenomena occur in an unprejudiced analysis of biblical geogra-
phy [1]. That many biblical texts describe volcanic activity has been stressed
in history long ago.
“The Lord said to Moses, ‘I am now coming to you in a thick cloud. . . . But when
the ram’s horn sounds (when the cloud leaves Mount Sinai — A.F.), they may go up
the mountain’ . . . there were peals of thunder and flashes of lightning, a dense cloud
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on the mountain and a loud trumpet blast. . . . Mount Sinai was all smoking because
the Lord had come down upon it in fire; the smoke went up like the smoke of a kiln
. . . and the sound of the trumpet grew ever louder” (Ex 19:9, 13, 16, 18).
“. . . all the people saw how it thundered and the lightning flashed, when they heard
the trumpet sound and saw the mountain smoking ...” (Ex 20:18);
u . . . you stood ... at Horeb. . . . The mountain was ablaze with fire to the very
skies: there was darkness, cloud, and thick mist” (Dt:10-ll).
The destruction of Sodom and Gomorrah has long been regarded in history
to have been due to a volcanic eruption. For example:
M . . . and then the Lord rained down fire and brimstone from the skies on Sodom
and Gomorrah. ... He saw thick smoke rising high from the earth like the smoke of a
lime-kiln” (Gn 19:24,28).
Here is a list of apparent volcanic eruptions mentioned in the Bible (com-
piled by V.P. Fomenko and T.G. Fomenko): Gn 19:18,24; Ex 13:21, 22; 14:18;
20:15; 24:15,16,17; Nm 14:14; 21:28; 26:10; Dt 4:11,36; 5:19,20,21; 9:15,21;
10:4; 32:22; S 22:8-10,13; 1 K 18:38,39; 19:11,12; 2 K 1:10-12,14; Ne 9:12,19;
Ps 11:6; Ps 106:17; Ps 106:18; Ezk 38:22; Je 40, 8:45, Lam 2:3; 4:11; Is 4:5;
5:25; 9:17,18; 10:17; 30:30; J1 2:3,5,10.
To associate (as is done traditionally) all these descriptions with Mt. Sinai
(and Jerusalem) seems doubtful; it is generally known that it has never been
a volcano.
Where did the events occur then? It suffices to study the geological map of
the Mediterranean area ([61], pp. 380-381, 461). There are no acting volcanoes
in the Sinai peninsula, Syria, or Palestine; there are only zones of tertiary and
quaternary volcanism, as, for example, near Paris. In the above-mentioned
regions, where the biblical events are traditionally located, no volcanic activity
has been discovered since the birth of Christ.
The only powerful, and by the way, acting volcanic zone, is Italy together
with Sicily. Besides, Egypt and North Africa have no volcanoes [61]. Thus, we
have to find
(1) a powerful volcano active in the historical era;
(2) a destroyed capital (see the book of the Prophet Jeremiah) near the
volcano;
(3) two other cities destroyed by the volcano, namely, Sodom and Gomorrah.
There exists such a volcano in the Mediterranean, and it is unique, namely,
the famous Vesuvius, one of the most powerful volcanoes in history. Famed
Pompeii (“capital”?) and two destroyed cities Stabiae (Sodom?) and Hercu-
laneum (Gomorrah?) are located nearby. We cannot but mention a certain
similarity in the names of these Italian and biblical towns. It is possible that
the name of Sinai for Vesuvius originates from the Latin Sino (sinus), and
Horeb from the Latin horribilis (horrible).
The following analytic study worth mentioning, which permits to read the
vowel- free text of the Bible, was performed in [1]. It took into account placing
Mt. Sinai-Horeb-Zion in Italy. We illustrate by examples.
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109
“The Lord our God spoke to us at Horeb and said, ‘You have stayed on this moun-
tain long enough; go now, make for all Canaan. . . ,M (Dt 1:7).
The theologicans supply the Hebrew KNN with vowels as Canaan and place
it in the desert on the Dead Sea coast, but another solution is also possible,
namely, KNN = GENUA (Italian Genoa), “. . . all Canaan and the Lebanon
...” (Dt 1:7).
The theologicans restore the Hebrew LBN with vowels as Lebanon; however
lebdnon means “white” , i.e., the same as Mont Blanc, or White Mountain: “. . .
as far as the great river, the PRT” (Dt 1:7). The theologicans restore PRT
with vowels and decipher it as the Euphrates; but, there is the large tributary
of the Danube, the Prut, located in central Europe, as beyond Mont Blanc.
“Then we set out from Horeb . . . and marched through that vast and terrible
wilderness” (Dt 1:19).
In fact, the famous Phlegraei, vast and burnt-out spaces filled with small
volcanoes, fumaroles, and solidified lava streams are located near Vesuvius-
Horeb. “. . . and so we came to KDS-BRN” (Dt 1:19). KDS-BRN is supplied
with vowels as Kadesh-Barnea, which is, possibly, a town on the Rhone ([1],
Vol. 2, p. 166). It is also possible that modern Geneva was meant: “. . . and
we spent many days marching round the hill-country of Seir” (Dt 2:1). Mount
Seir was left without translation; however, if it is translated, we obtain Devil’s
Mountain(s). And there is such a mountain near Lake Geneva, namely Le
Diableret (“Devil’s Mountain”). Then, the “Children of Lot” (Dt 2:9) met on
the way can be identified with the Latins (LT) ([1], Vol. 2, p. 167); “. . . and
cross the gorge of the Arnon ...” (Dt 2:24); but, this is the Italian river Arno:
“Next we ... advanced ... to Bashan” (Dt 3:1). The town Bashan is often
mentioned in the Bible. It is surprising that Bassano still exists in Lombardy.
“. . . king of Bashan ... came out against us at Edrei” (Dt 3:1). Adria is
still here, on the Po delta; the Po, by the way, has often been mentioned by
ancient Latin authors (e.g., Procopius) and called the Jordan (in Procopius’
Eridanus) i which is very consistent with the biblical spelling of the Jordan,
namely, hay-yarden ([1], Vol. 2, p. 167). .. and we captured all his cities
. . . sixty cities ...” (Dt 3:3-4). Indeed, in the Middle Ages, there were many
big cities in the region: Verona, Padua, Ferrara, Bologna, and others. “. . .
from the gorge of the Arnon to Mount Hermon (= HRMN)” (Dt 3:8). But
it is obvious that MNT HRMN can be supplied with vowels to be translated
as the “German mountains”. “Only the Og king of Bashan remained . . . His
sarcophagus of iron may still be seen in the . . . city of Rabbah” (Dt 3:11).
Here is mentioned not only Ravenna, but also the famous tomb of Theodoric
of the Ostrogoths (Og = Goths?), built in A.D. 493-526. There follows TBRH
(Taberah in biblical translation), which is naturally identified with the Tiber
in Italy; ZN is Siena, southeast of Livorno; and HBRN, i.e. “Gorge du Rhone”,
a town on the Rhone ([1], Vol. 2, pp. 229-237). The slopes of Monte Viso are
called Jebus (Jgs 19:10-11) in the Bible, and Rome is called Ramah (Jgs
19:14).
110 New Experimental and Statistical Methods Chapter 3
According to the statistical analysis of ancient texts (see details below),
and if the vowels are left out, the following geographical identifications may
be consistent: Assyria = Germany, France = Persia (PRS), Jerusalem = Rome
or Pompeii, Media = Hungary. Furthermore, due to the impossibility of the
European location of many of the biblical events and terms, we stress that
the word Venetiae could have been read by the ancients both as Venice and
Phoenicia ( Phoinike ).
8.14. Ancient originals and medieval duplicates. Anachronisms as a
common feature in medieval chronicles
The “Renaissance effect” of duplicating antiquity is vividly expressed in tra-
ditional chronology.
Plato is the founder of Platonism; his teachings were revived several hun-
dreds of years later by another, the famous Neoplatonist Plotinus (a.d. 205-
270). His name is almost identical to that of his teacher Plato. Neoplatonism
then died, too, but was revived again in the 11th century by another famous
Platonist, Pletho, a name which is again accidentally almost identical to Plato.
Freed of all vowels, Plato, Plotinus, and Pletho are simply identical names.
It is assumed that Pletho revived ancient Platonism. Plato’s manuscripts
first appeared from oblivion just in the time of Pletho, i.e., in the 15th cen-
tury [62]. Pletho organized the well-known Pletho Academy in Florence, an
exact analogue of the ancient Platonic Academy [62]. He was the author of
the famous utopia (both Plato and Pletho wrote utopias), the Laws , which,
unfortunately, has not survived in complete form. But the text of Plato’s Laws
was preserved. Like Plato, the 15th-century Pletho put forward the idea of
an ideal state, his theory being extremely close to that of Plato. “Imitating”
them both, Plotinus also hoped that the emperor would help him found the
town of Platonople in Campania (again in Italy), where he would introduce
aristocratic and communal institutions “conceived by Plato” ([63], Vol. 4).
The number of these strange “duplicates” in traditional chronology is quite
large.
One of the basic reasons for at least two variants of dating ancient and
“medieval” documents is due to the Renaissance, when all the antique (now
regarded as ancient) branches of science, philosophy, culture, painting, etc.,
were revived. It is assumed that the “ancient and brilliant Latin language” de-
graded at the beginning of the Middle Ages into a rude and awkward tongue
which started acquiring, and did acquire, its former brilliancy only during
the Renaissance. This “Renaissance” of the Latin language (as also that of
ancient Greek) did not start any earlier than the 8th-9th century [64]. Start-
ing with the 10th— 1 1th century, the famous medieval trouveres came to make
use of the stories now called by the historians “the masquerade of classical
remembrances” [64]. In the 11th century, The History of Ulysses appeared,
in which the well-known (and allegedly Homer’s) fabula was retold from the
“medieval approach” (speaking of knights, ladies, tournaments, etc.); but on
the other hand, all the elements which would later be regarded as intrinsic
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Problems of Historical Chronology
111
of any antique story had already been present ([64]; see also the “antique”
activity of the famous medieval poet Homer (Angilbert), who lived at Charle-
magne’s court in the 9th century. He was “the most important member of
the scientific community at the Aachen court” ([63], Vol. 5). According to
J. Demogeot:
“It was the end of the 12th or in the 13th century when . . . the trouveres . . .
started to speak with a certain satisfaction, ‘this story (i.e., the story of the Trojan
War — A.F.) is not commonplace; nobody has yet written it*. For them, it was almost
a traditional story” ([64], p. 110). 3
As a matter of fact, the Franks regarded themselves as formerly having lived
in Troy (!), whereas one author in the 7th century, Fredegarius Scholasticus,
points out king Priam as a figure belonging to the medieval epoch [64].
The Argonauts’ expedition was also claimed to have coincided with the Tro-
jan War when the conquering Crusaders, analogues of the Argonauts, rushed
into the far regions of Asia. Alexander the Great flooded France with com-
pliments [64]. Speaking of the Trojan war, some medieval authors call Paris
(Alexandros) a Parisian ([57], Comm. 76, p. 234) - French name.
The medieval authors believed that the emperor Heraclius (traditionally
assumed to have reigned from 610 to 641) had reigned with Princess Semlramis
in the Land of the Hellenes ([57], p. 107). If we place biblical Semlramis into
the 7th century, we shall diverge from traditional chronology to the extent of
several hundreds of years.
A.V. Sterligov, the author of Ancient Plots in French Book Illustration at
the End of the 14th-15th century [91], for example, reports the following:
“The greatest specialist in French medieval literature Gaston Paris wrote that the
Middle Ages had never recognized themselves how far they had been separated fromm
antiquity. . . ”
“Anachronism was a common feature in the numerous universal chronicles, and an-
cient historians’ renderings where biblical; ancient and national historical characters
and the contemporaries were depicted similarly. The Trojan heroes cannot be distin-
guished from King Arthur’s knights and warriors of the Hundred Years* War, while
the contemporaries of Charles VI pronounce phrases which Livy put in the mouth of
Gaius Capuleius, a tribune living in the 5th century B.C. ... Ancient history, espe-
cially that of Troy, from which the French nation allegedly originated, was included in
the French chronicles. Hence, say, during the famous 1330 tournament, the Parisians
fought under the name of Priam and his 35 sons” ([91], p. 236).
“The deeds of biblical, ancient Greek and Roman heroes merged into one history
of knighthood, continuing before the eyes of contemporaries” ([91], p. 237).
Under the pressure of tradition and all these strange things, the historians
are forced to believe that
“. . . the medieval idea of chronological sequence had all but been confused: Monks
with crosses and censers took part in the funeral of Alexander the Great; Catiline
3 Translated from the French — tr.
112
New Experimental and Statistical Methods
Chapter 3
in the liturgy . . . Orpheus was in their eyes Aeneas' contemporary, Sardanapalus a
Greek king, Julian the Apostate a papal chaplain. All in that world took fantastic
colouring according to the modem historians.. . . The most glaring anachronisms and
the strangest fantasies neighboured peacefully” ([113], pp. 237-238).
All the above facts (and thousands of others!) today have been rejected as
“preposterous”. However, we have to bear in mind that their seeming prepos-
terousness only arises from the chronology traditionally accepted at present.
Long before the discovery of the allegedly “ancient” manuscript of the story
of the Golden Ass, the “ass theme” had been widely exploited by medieval
trou veres, and the “ancient” story (of Apuleius), which surfaced only during
the Renaissance, resulted from the medieval sources. It is generally known that
in the Middle Ages, much earlier than the “ancient” originals were discovered,
all “ancient” stories had been exploited and embellished, with the alleged orig-
inals chronologically and evolutionally following their medieval predecessors
[64].
The oldest biography of Aristotle is dated to a.d. 1300 ([86], p. 29). Further-
more, only 20 percent of what is now called the Aristotelian corpus belongs
to Aristotle himself ([86], p. 64). In the 15th century, it was the propagandist
Georgios Scholarios who fervently popularized Aristotle’s works.
8.15. Names and nicknames. Handwritten books
It is important that people in antiquity did not have names in the contem-
porary sense of the word but only had nicknames meaningful in the original
tongue. The father of a Roman consul in 169 B.c. had 13 names, and his
son 38. The nicknames characterized a man; the more remarkable traits of
character, the more nicknames. Different chroniclers gave an emperor the dif-
ferent nicknames under which he was known. Moreover, they sometimes knew
the same emperor under different nicknames. Pharaohs bore one name before
the coronation, and another after it. Since they were crowned several times
in different regions, the number of “names” grew sharply. For example, these
nicknames were strong, light, and so forth.
“The csar Ivan III had the name Timothy; Vassily III was Gabriel . . . ; the csarevitch
Dmitry murdered in Uglich was Uar. One name was regal, and the other Christian”
([98], p. 22).
Traditional history believes that medieval names were different from an-
cient ones. However, textual analysis demonstrates that ancient names were
quite often used in the Middle Ages. Nilus Ancyranus (who died in A.D. 450)
wrote letters to the contemporary monks Apollodorus, Amphictyon, Atticus,
Anaxagoras, Demophanes, Asclepiades, Aristocles, Aristarchus, Alcibiades,
Antiochus, Apollos, and others [99]. An exceptionally large number of ancient
names were used in the 12th- to 14th-century Byzantine Empire.
Handwritten books have long survived the beginning of printing, success-
fully competing with the latter throughout Europe ([94], pp. 19, 25). With
a few exceptions, almost all Irish literature of the 7th-8th century “only ex-
ists in manuscript form” ([94], p. 28). Until 1500, 77 percent of all printed
§9
Astronomical and Mathematical Analysis of the Almagest
113
books were in Latin, since the Latin type was easier to make. However, the
types of other languages were being introduced into the printing practice ex-
tremely slowly, because the technology of making special marks to denote
accents, vowels, and so forth, was complicated. Therefore, for many years af-
ter printed books had appeared, “people copying Greek, Arabic and Jewish
manuscripts did not remain unemployed — ” ([94], p. 57). Especially many
hand-written books of the book-printing era were made in Greece (!). It is
possible that many of these manuscripts were later declared “ancient” . All of
the above facts make many of the paleographic instances of dating doubtful.
It turns out that many manuscripts were copied from books already printed
([94], p. 120). Greek monasteries were especially famous for their scribes who
copied printed books! To detect such manuscripts, one should compare the
textual errors with those in printed editions. Most probably, in copying a
book, the errors were copied, too.
§9. Astronomical and Mathematical Analysis of the Almagest
9.1. Morozov’s analysis of the first medieval editions of the
Almagest
The basics of traditional chronology were established by analyzing written
sources. If we reconsider the datings, freeing them from the a priori accepted
hypotheses regarding the age of the documents, we shall not find any seri-
ous contradictions, the typical example being the new dating of Ptolemy’s
Almagest , carried out in [1]. Its most important part is the star catalogue:
The star coordinates are given with an accuracy of 1/6°. To obtain so much
accuracy, a clock with a minute hand is required; however, such a clock was
not invented until the 12th-15th century.
Morozov discovered that there was a reliable method to find out when the
catalogue was actually made. Without going into details, we know that the
stars’ longitudes possess an annual precession of 50.2"; hence, by dividing
the difference between today’s longitudes and those indicated by Ptolemy by
50.2", we derive at the year when the catalogue was compiled. The result is
shocking: All the longitudes given in the first Latin edition could have been
recorded in the 16th century, i.e. , when the book was published (now regarded
as “ancient”, and assigned to the 2nd century A.D.). This fact had not been
noticed earlier, since the astronomers studied the Greek edition. Allegedly,
the Greek text is the original one but was published later than the above-
mentioned Latin “translation”. The (second) Greek edition supplies values
for the longitudes, decreased by 20° ± 10', as if the stars had been observed
in the 2nd century a.d, which is the traditional dating for the Almagest.
A hypothesis arises that the Latin text was the original and the Greek the
secondary one, but not vice versa as in the traditional approach. But it is also
possible that, in the 16th century, the Almagest was published as a treatise
for practical use and not as a historical document; and since the data had
114 New Experimental and Statistical Methods Chapter 3
become obsolete due to the precession and was of no purpose, the translator
“refreshed” the catalogue by introducing the latest evidence.
This objection can be eliminated by taking notice of the fact that the most
important star coordinates were improved considerably in the later Greek
edition, compared with the Latin one. Thus, by “restoring” Ptolemy’s data
as far as precession goes, the editor also improved it in other respects, which
is not consistent with the conjecture that the Greek text was the original.
It was noted in [1] that the following about the Almagest seemed to be very
strange:
(1) The extraordinary state of preservation of the star catalogue and the
whole text. The Almagest is traditionally believed to have been rewritten
many times since the 2nd century a.d. But, in copying, numerous errors must
have been made, which is not confirmed, however.
(2) The inexplicable accuracy of the catalogue and other observational data
collected by Ptolemy and Hipparchus.
(3) The use of the spring equinox as the basic point for measuring the
longitudes, which could only be done by overcoming certain difficulties (one
started resorting to it only since the 10th-16th century).
(4) Dating the Latin edition (from the longitudinal precession) to the 16th
century, agreeing with the date of the 2nd century A.D. (more precisely, by
the rule of Antoninus Pius) given in the Almagest itself, is confirmed by the
Greek text only if we take the value of the 16th-century precession, and is
rejected if we replace it by a more precise one.
(5) The “improvement” of the star catalogue of the Greek edition consisting
of (a) more precise star coordinates, and (b) a systematic correction of their
latitudes, obviously due to taking refraction into account (a late medieval
discovery).
(6) The systematic and considerable latitudinal shift discovered by the as-
tronomer J. Bode, which puts any explanation of the Almagest's traditional
date to doubt.
(7) The choice of the North Star as the first star of the catalogue, which nei-
ther could have been substantiated astronomically in the 2nd century (when
another star was near the north galactic pole) nor have been consistent with
the ecliptical coordinate system adopted in the Almagest , but quite under-
standable if the observations were made in the 1 1th— 16th century.
(8) The inclusion of the star Achernar, which could not have been seen in
Alexandria in the 2nd century A.D., but was already visible in the 15th-16th
century.
(9) The inclusion of Albrecht Diirer’s engravings (astrographic charts) made
only in 1515. As a matter of fact, many stars were localized on the map relative
to these figures as being “in the leg of the constellation Pegasus”, and so forth.
In my opinion, all these remarks made in [1] are very subjective and can be
interpreted in different ways. We need some strong statistical method for dat-
ing the star catalogue. The difference between the Latin and Greek editions of
the Almagest (see items 4 and 5) shows that we need to base our research on
§9
Astronomical and Mathematical Analysis of the Almagest
115
the ancient manuscripts of the Almagest. We base our work on the “canoni-
cal” version of the Almagest's star catalogue as presented in the fundamental
work of Peters and Knobel ( Ptolemy's Catalogue of Stars. A revision of the
Almagest , Carnegie Institute, Washington, 1915). They investigated all the
main manuscripts of the star catalogue.
The attempts to date the catalogue on the basis of the displacement of
individual stars such as Arcturus are not convincing. This motion is extremely
slow, only several seconds a year. Recall that the astronomer Halley, on the
contrary, calculated the value of Arcturus’ proper motion, referring to star
longitudes and the traditional dating of the Almagest to the 2nd century A.D.
In reality, the values of the proper star displacements are small and of the
order of the measuring errors; therefore, to date the catalogue reliably on
the basis of individual displacements is complicated. (See details in the next
section.)
Dating catalogues by the deviation from modern star coordinates should be
generally very accurate. For example, it is on the basis of a ca. 20° difference
between the longitudes in the Greek and the Latin editions that traditional
chronology makes a conclusion about their having been observed in the 2nd
century A.D. It is tacitly assumed, however, that both the ancient and me-
dieval astronomers counted longitudes from the same “basic point”. To give
just one example, the famous Theatrum Cometicum by the 17th-century au-
thor Stanislaw Lubieniecki should be “dated” to the 5th century B.C.! This
absurd conclusion is due to Lubieniecki’s having counted the longitudes off
the first stars of Taurus, which led to a large difference with the modern
longitudes, and which is not accepted today. There are very many similar
examples [1]. Methods of the computation of longitudes simply were diverse
in the Middle Ages, and each author took different stars as initial reference
points, resulting in a difference in the longitudes, and therefore not at all
explained by precession.
9.2. On the statistical characteristics of the Almagest. The
structure of the star catalogue
Sections 2-5 contain the results of a 1986 work by the author and G.V. No-
sovsky:
The history of the Almagest is well known. It is assumed traditionally that
its star catalogue was made by Ptolemy in A.D. 138 from his own observations
in Alexandria, though the fact that the stars were observed by Ptolemy himself
was denied by many researchers (e.g., in [102]). R. Newton’s book presents
some of the latest and quite fundamental research on the Almagest. Therefore,
we only touch upon the history of the discovery of the catalogue in medieval
Europe.
The catalogue’s manuscript got into Rome from Byzantium in ca. 1450 (re-
call which 1453 is the year when Constantinople fell). It is assumed which its
first translation into Latin was made by Georgios Trapezuntius in 1460 and
that it was published with Diirer’s maps only in 1528 in Venice (see [108],
116
New Experimental and Statistical Methods
Chapter 3
pp. 10-11). There are data indicating that the famous scientist and his con-
temporary, I. Regiomontanus, investigated the catalogue in 1464, which was
published only in 1541 ([108], p. 220). Here is the list of the first editions of
the Almagest : 1496 in Venice (?), abbreviated by Regiomontanus and G. Pur-
bachi ([108], p. 218); 1515 in Venice, Liechtenstein’s edition in Latin; 1528, in
Venice, Trapezuntius’ edition in Latin, with the star longitudes corresponding
to those of the 16th century; 1537, in Basel, Trapezuntius’ edition in Greek,
which he called the “first” edition, with the ancient star longitudes.
The catalogue’s structure was as follows: Stars are divided as in the con-
stellations, each description consisting of four columns with the positions de-
scribed as, for example, “in Pegasus’ head”, ecliptical longitudes, latitudes
and magnitudes (with brightness being in inverse proportion to the magni-
tude). Sometimes, the description of a constellation is followed by that of the
stars near it, but which are not in the corresponding figure. Today, all the
stars are formally included into the corresponding constellations.
We now discuss each of the columns.
The position of a star in a constellation. This column is least infor-
mative. As a matter of fact, neither constellations nor the descriptions of the
stars’ positions had ever been standardized in the Middle Ages. In the 19th
century, they were understood as simply being a certain region in the astro-
graphic charts under a constellation; however, it was essential for the medieval
astronomers that the stars should be well consistent with a particular figure.
The forms of the constellations in medieval astronomical textbooks seem fan-
tastic, and it is hard to understand which particular configuration of stars was
meant (see examples in [1]). The first standard representations of the sky were
given in Diirer’s prints, published together with the Almagest in 1537, though
they were not completely identified with the modern star atlas. The descrip-
tion of the stars in the constellations did not become standard after Diirer’s
maps either: The Almagest , Copernicus’, and Tycho Brahe’s catalogues are
all different. It is only in the 17th century that the first sufficiently precise
classification systems appeared, which made it possible to identify with cer-
tainty the stars mentioned in ancient catalogues with modern stars (i.e., with
stars listed in modern catalogues.
Star coordinates. Two star coordinate systems were known in the Middle
Ages, namely, ecliptical and equatorial systems. From now on, we refer the
reader to [102] with respect to all problems of star astronomy. Beginning with
the 17th century, only equatorial coordinates have been used in catalogues.
Ecliptical coordinates are only seen in the Almagest and in 16th-century cat-
alogues, the last of them being by Tycho Brahe. The reason for this is very
simple: Equatorial coordinates are much easier to measure (to a greater accu-
racy at that) than ecliptical ones. On the other hand, at the time of Ptolemy
and from the 15th to the 17th century, astronomers believed ecliptical coor-
dinates to be “eternal”, i.e., suitable at once for all epochs, with the latitudes
being invariable in time, and the star longitudes varying at a constant rate
§9
Astronomical and Mathematical Analysis of the Almagest
117
(meaning the longitudinal precession). Therefore, the two principal questions
discussed by astronomy in the 15— 17th century were: Do stars possess their
own movement, or is the configuration of the starry sky invariable, and what
is the rate of precession? Both answers given by Ptolemy are wrong. He an-
swered in the negative in the first case, though Arcturus’ own movement was
discovered in the mid-18th century. He determined the rate as 1° per 100
years, its modern value being 49.8s a year, i.e., 1.4° each 100 years. It is cu-
rious to see Tycho Brahe assert that Copernicus allegedly assumed that the
precession was 1° a century, and Tycho Brahe ascribed the first refining of
the figure to himself (see [104], pp. 377-396).
The division value for star coordinates in the Almagest is 10 minutes. More
precisely, one-sixth of a degree, since the Almagest coordinates were given in
degrees and their fractions with an extremely inconvenient (from the modern
standpoint) method for the latter notation (e.g., see [102], [108]). The coor-
dinate accuracy does not correspond to this value (see the analysis in [102]
and in the following). Ptolemy declared that he had directly measured the
ecliptical coordinates of all stars by means of an astrolabe, which is a rather
complicated procedure [102]. Copernicus only refined the positions of several
stars. Tycho Brahe rewrote the catalogue again, using the system of base stars
and thus achieving the accuracy of one minute. His catalogue contains about
700 stars (against Ptolemy’s 1022); however, after it had been published in
the 16th century, the Almagest completely lost its scientific importance.
Magnitude of a star. The scale of magnitudes in Ptolemy’s Almagest is
made up of integers from unity (corresponding to the brightest stars) to six
(corresponding to the faintest ones). He stated that he had observed all stars
up to the sixth magnitude, which seems improbable from today’s point of
view, since many stars of magnitude 3 to 5 cannot be identified with them,
whereas some of magnitude 6 can be identified at once. If we compare the
magnitudes of Ptolemy’s stars with the modern ones, we shall see that the
magnitudes 1 to 2 were always determined correctly, whereas he often made
mistakes in the interval of the magnitudes 3 to 6.
We now go on to the problem of dating the observations of the stars which
formed the basis for the Almagest star catalogue. Its traditional dating is
founded on Ptolemy’s statement that the catalogue had been compiled in the
second year of the rule of the emperor Pius. According to the Scaliger-Petavius
chronology, this is the year A.D. 138, though the name “Pius” was common to
some other Roman emperors, for example, the famous Maximilian I Pius (1493-
1519), who reigned just at the time of the first Almagest editions (Fig. 9(2)).
However, even Delambre noticed that the average error in the star longitudes
was about 35 minutes; therefore, the catalogue supplies the coordinates not as
they were in A.D. 138 but in ca. in A.D. 60. J . Bode speaks of the year 63, whereas
F. Peters and E. Knobel of the year 58. In fact, the error variance for the lon-
gitudes of approximately 1,000 stars is about 40 2 minutes. Hence, the average
error in longitude is a random variable with variance a 2 = 40 2 /1000 = 1.6'.
118
New Experimental and Statistical Methods
Chapter 3
Since they must be distributed according to JV(0, 1), the admissible average
error in the longitude should not exceed 5 minutes, which permits us to de-
termine the epoch to which the catalogue dates to the accuracy of 5-6 years.
Thus, in the form it was published in Basel in 1537, the Almagest could not
have been made from the observational data of A.D. 138. This last important
remark by Delambre led to the hypothesis that Ptolemy had not observed the
stars by himself, but rather updated the Hipparchus catalogue to A.D. 138, the
latter being dated to 130 B.c. Meanwhile, making use of the incorrect preces-
sion value, he actually reduced the catalogue to A.D. 60 instead of A.D. 138.
The authors of [105], [106], and [108] were of the same opinion. The important
work [102] is devoted just to the proof of this hypothesis. Note that almost all
of its conclusions are based on the assumption that the Almagest was written
by an ancient astronomer. However, R. Newton’s main argument based on the
comparison of degree fractional part distribution for latitude and longitude of
a star is not related to chronology [102]. In this connection, the investigation of
the distributions of degree fractional parts in Ptolemy’s catalogue, carried out
by Newton, only permits us to state the fact that the catalogue was obtained
from the original and is based on direct observations by adding an integral
(possibly, negative!) number of one degree and forty minutes.
If, following [108], we assume that the Almagest is related to A.D. 58, and if
we take precession into account, then the longitudinal shift through an integral
multiple of one degree and forty minutes takes us to the following values for
the epochs of observation, namely, ..., —207, —134, —62, 10, 92, 154, 227,
299, 371, 443, 515, 586, 660, 732, 805, 877, 950, 1022, 1094, 1166, 1239, 1311,
1383, 1456, 1528, 1600, . . . (years B.c. are indicated by a minus sign).
The conclusion of Newton is that the Ptolemy star catalogue had been
“corrected” according to the precession used by Hipparchus. On the basis
of statistical investigations, he asserts that Ptolemy was a falsifier. The same
thought was even expressed in the book’s title, namely, The Crime of Claudius
Ptolemy , and in the titles of many of the book’s sections. For example, he
speaks of equinoxes and solstices allegedly observed by Ptolemy, the fabricated
solstice of 431 B.C., observations allegedly made by Ptolemy to determine
the ecliptic’s slope and Alexandria’s latitude, four fabricated lunar eclipse
“triads”, the proof of a forgery, a swindler, falsifications of calculations and
falsifications with oversights, falsification of data, falsification of the Venus
data and exterior planet data, and so forth.
Newton writes that all the observations made by Ptolemy himself, so far as
they can be checked, turned out to be a forgery, and that many observations
ascribed to other astronomers were also part of the falsification. It then be-
comes clear that none of Ptolemy’s statements can be accepted unless they
have been confirmed by authors absolutely independent of his data. Be it his-
tory or astronomy, all research based on the Almagest should be performed
again. Thus, Newton completes his thought, Ptolemy is not the outstanding
astronomer of antiquity but a more unusual figure: a most successful swindler
in the history of science (see [102]).
§9
Astronomical and Mathematical Analysis of the Almagest
119
However, this estimation of Ptolemy’s work by a well-known astronomer of
modern times is only based on the hypothesis of the Almagest's ancient origin.
The attitude toward Ptolemy may become quite different if it turns out that
we are dealing with a text written in the 10-16th century. We have already
noted earlier that almost all of Newton’s conclusions depend on the a priori
dating of the catalogue to about the year A.D. 1, or the first two centuries
before and after. Our goal is the Almagest s analysis based solely on the star
catalogue itself and the contemporary idea of the starry sky. Some steps in
this direction are made in the sequel. We once again stress the importance of
Newton’s research and would only like to complete it by suggesting another
interpretation.
Below, we have collected the data regarding the errors in the star catalogue.
9.3. The accuracy of the Almagest’s star coordinates
Here, we discuss the accuracy of the Almagest's star coordinates and the way
Ptolemy measured them. Recall that the division value was 10 arc min., or
one-sixth degree. Much work on identifying the stars in the Almagest catalogue
with the stars of modern time was done by Bayer, Flamsteed, Bode, Baily, Pe-
ters, Knobel, and others in the 16-20th century. This was a very nontrivial job
(for details, see [108], which gives a large number of versions of identification,
with different authors identifying the same stars differently in the different
Almagest manuscripts). Since not all the Almagest stars have been identified
with certainty, we only consider the zodiacal stars. The zodiacal constellations
were especially important for ancient and medieval astronomy and astrology.
It is, therefore, reasonable to assume that their coordinates were measured
more often than the coordinates of other stars. Hence, the coordinates of the
zodiacal stars apparently make a homogeneous sample, whereas the sample
of all the Almagest star coordinates is inhomogeneous, in particular, due to
the different accuracy of determining longitudes in different latitudes (how-
ever, there may be other reasons, too; e.g., astrological, more important stars
probably having been measured in a more thorough manner).
The Almagest's zodiac consists of 350 stars. The average error in longitude
is zero if A.D. 60 is taken as the observation epoch (more precisely, this is the
epoch to which the catalogue is related, with it possibly being much different
from the time when the actual observations were made; see above). The aver-
age error in latitude is —2.4°, the sample variance in the longitudes <7 2 = 47 2
min. (recall that 1°= 60 min.). Ten values exceeding 130 minutes were not
taken into account. The sample variance in latitude was <7 2 = 26 2 min., with
one value of 136 minutes being rejected. We can draw two conclusions from
comparing these and the division value of 10 minutes in the Almagest.
Apparently, the longitudes were recalculated after actual observations. In
fact, since the accuracy of the longitudes is much worse than that of the lat-
itudes (remember that this difference cannot be due to the proximity of the
stars to the poles, considering only the zodiacal stars with latitudes within
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New Experimental and Statistical Methods
Chapter 3
20°), this circumstance is another argument for the recalculation of the longi-
tude. In antiquity and the Middle Ages, calculations involved complex literal
designations of integers and especially fractional parts, constructed from frac-
tional units denoted in the same way as their denominator, but primed (see
[102], [105]). It is natural that, in each recalculation, errors arose and the
accuracy fell.
The variances of the catalogue errors are not consistent with the division
value of 10 minutes. It turns out that, as a rule, Ptolemy’s mistakes consisted
of several divisions at once, which confirms Morozov’s standpoint that the
Almagest coordinates were obtained by a method quite usual for the Middle
Ages and, possibly, for antiquity, namely, by measuring equatorial coordinates
and subsequently recalculating them into ecliptical ones, done graphically on
large atlases or special terrestrial globes with grids of both coordinates (see [1],
Vol. 4, pp. 201, 264-265). However, it is assumed traditionally that Ptolemy
observed ecliptical coordinates directly by means of a complex device, the
astrolabe, adjusting it by the sun in the afternoon, and by calculating the
correction for the shift of its plane due to the earth’s rotation by a clepsydra.
If we accept Morozov’s viewpoint, then we derive at a natural and simple
account for these “strange” average error graphs in determining the zodiacal
star coordinates as the longitudinal function (stars are grouped with respect
to their longitude, and the average error in longitude and latitude is calculated
in each group; see the graphs in [108]). The graphs resemble two sine curves
shifted with respect to each other approximately by one-fourth of the period.
The fact can be easily explained by a small error (within 0.5°) in specifying the
ecliptic’s slope to the equator on the terrestrial globe (see also the discussion
in [1], Vol. 4).
Thus, the coordinates of the Almagest stars are, most probably, a total
of the following: (1) Measuring equatorial star coordinates, (2) recalculating
equatorial into ecliptical coordinates graphically, and (3) adding a certain
constant difference to the longitudes without altering the latitudes (possibly,
it was done several times). In connection to the latter item, we once again
stress the important investigation by Newton of the degree fractional parts
distribution for the catalogue coordinates, showing that an integral multiple
(possibly negative) of degrees and 40 minutes was added to the original lon-
gitudinal values (see [102]). This operation could have made the catalogue
more ancient and shift it from the 10— 15th century toward the turn of the
first millennium.
9.4. The problem of dating the Almagest from the individual stars’
proper motion
We now dwell on the problem of dating the Almagest based on the individual
stars’ proper motion. Similar hypotheses were repeatedly made. We distin-
guished all those stars from the Almagest whose own motion is greater than
one second a year, at least with respect to one of the coordinates 01900 , 61900,
where a and 6 are equatorial coordinates, right ascension and declination. The
§9
Astronomical and Mathematical Analysis of the Almagest
121
subscript indicates the epoch in which the coordinates were measured, i.e., the
year in which the spring equinox was the origin of the coordinates. Note that
the star’s own motion in a historical period can be regarded as rectilinear and
uniform.
It turned out that there were nine such stars: a in the constellation Bootes
(= Arcturus), a in Canis Major (= Sirius), a in Canis Minor (= Procyon),
1 of Perseus, r of Cetus, O 2 of Eridanus, 7 of Serpens, ij of Cassiopeia, and
a of Centaurus, the latter having been rejected because it is too southern (it
is hard to observe southern stars, because they do not rise high enough over
the horizon in the Northern Hemisphere), and it is given in the Almagest with
an error greater than 5°, which corresponds to its own motion in 5,000 years!
Thus, eight stars remained, the first three of them being of magnitude one;
therefore, their identification leaves no doubt, which cannot be said of the
other five, since a convenient star from the Almagest can only be selected on
the basis of the similarity of the star coordinates. In other words, we have to
take the nearest Almagest stars.
These stars turned out to be actually identified with the modern ones, only
because they were the nearest stars for certain ones from the catalogue and
are visible by the naked eye. We shall make this important thought more
precise. The 17th- to 18th-century astronomers were forced to identify many
Almagest stars just in this way. Knowing their own motion, they calculated
their position, or coordinates, with respect to the turn of the first millennium,
and then looked for the nearest stars in the Almagest Thus, the Almagest stars
got their names. It is obvious that the hypothesis about the Almagest being a
document dated to the start of the Christian era resides in the identification
procedure itself. The rejection of this a priori hypothesis can lead us to quite
different identifications. Finally, the star identification procedure on the basis
of the coordinate similarity, first substantially depends on the a priori cata-
logue dating and, second, is not at all unique (e.g., in the case where at once
several stars are equally eligible in position and magnitude). We stress once
again that, since we here consider the stars being displaced noticeably with
respect to a fixed star net, to compare them with the Ptolemy catalogue, we
must know the time of the observations. Thus, the star O 2 in Eridanus was
identified differently by different researchers (see [108] for the table of different
identifications). Moreover, the star with which it is now traditional to identify
0 2 in Eridanus (No. 779 in the Baily numeration, with Ptolemy not having
given a consecutive numeration) has several versions. The thing is that differ-
ent printed books and manuscripts supply different values for its coordinates.
We consider the two basic values due to Baily (see [106], chosen on the basis
of investigating printed books) and due to F. Peters and E. Knobel (see [108],
chosen on the basis of the Almagest manuscripts). The situation is similar
to the coordinates of Procyon and its identification, existing in two versions
of the printed books, and of Baily (see [106]), and in some manuscripts and
[108] -
Thus, the question “who is who” in the Ptolemy catalogue is absolutely
122 New Experimental and Statistical Methods Chapter 3
nontrivial for some stars, and its solution is substantially dependent on dating
the Almagest a priori .
A point estimation of the time when it was observed by the least-squares
methods was found for each of the considered stars. Namely, the perpendicular
was dropped from the position of an Almagest star to the trajectory of the
corresponding star's own motion in today’s sky, and the year was found when
the moving star was at the foot of this perpendicular, for which ecliptical
coordinates were recalculated into equatorial ones, and reduced to the 1900
equinox. Meanwhile, the Almagest epoch was taken as A.D. 60 (see [105],
[106]).
We now list the obtained estimates: (1) A.D. 800 for Arcturus, (2) a.d. 500
for Sirius, (3) A.D. 1850 for Procyon (according to Baily) or A.D. 1450 (ac-
cording to Peters and Knobel), (4) 2000 B.c. for i in Perseus, (5) 150 B.c. for
r in Cetus, (6) A.D. 1800 for O 2 in Eridanus (according to Baily) or 150 B.c.
(according to Peters and Knobel), (7) 1400 B.c. for y in Serpens, and (8) 100
B.C. for r) in Cassiopeia.
However, note that the roughest estimation of the accuracy of these dates is
600 years. It is obtained if the division value in the Almagest , i.e., 10 minutes,
or 600 seconds, is divided by the rate of the star’s own motion, or about 1
second a year. But, if we recall the Almagest star coordinate error variance
(see above), then this value should be increased at least twice, though the low
accuracy of the obtained estimates can already be seen from their scattering.
The trajectories of the above eight stars own motion in the equatorial co-
ordinates aigoo, ^1900 are represented in Fig. 20 (1-7) and 21, the positions
in 1900 are at the point where the axes meet. All bright stars (up to magni-
tude 6.5), falling into a given region, are also represented. The figure near a
star is its magnitude, and its number in the catalogue of [107] is also given.
Figures 20(5) and 20(6) explicitly demonstrate that the identifications of the
moving stars with those from the Almagest are substantially dependent on its
dating: If we assume that it was made, say, in the Middle Ages, then No. 779
is most probably associated with No. 1362 (or No. 1363) from the bright-star
catalogue [107], whereas No. 265 from the Almagest with No. 35k Ser (Nos. of
the Almagest stars are given according to Baily; see [106], [108]). Thus, dating
the Almagest from the own motion of individual stars with the use of their
traditional identification with the Ptolemy stars can still be false even if we
neglect the accuracy of the datings obtained, since the traditional identifica-
tions were made in the 18-20th century, involving the rate of the star’s own
motion and the Almagest f s a priori dating to the 1st century A.D., which can
be seen especially well in Fig. 20(5). We can find identifications corresponding
to the medieval, and not at all ancient, observation epoch for both versions of
the coordinates of No. 77 9 in the Almagest , which is in Eridanus according
to Baily, and No. 1362 (or 1463) from the bright-star catalogue according to
Peters and Knobel [107].
Our first conclusion is an inconsistency between the characteristics of the
star catalogue itself and the statements contained in Ptolemy’s astronomical
§9
Astronomical and Mathematical Analysis of the Almagest
123
Figure 20(1). The real motion of Arcturus and its position according to the
Almagest.
Figure 20(2). The real motion of Sirius and its position according to the Al-
magest.
treatise. The time of the observations according to Ptolemy, if dated tradition-
ally, the 2nd century A.D., does not correspond to the catalogue longitudes.
The descriptions of the observations themselves are extremely doubtful, since
124
New Experimental and Statistical Methods
Chapter 3
Procyon (Almagest)
✓ N
/ \
O' 10* 20* 30'
Figure 20(3). The real motion of Procyon and its position according to the
Almagest.
Figure 20(4). The real motion of the star i Per and the position of the star
196 according to the Almagest.
they are not consistent with the catalogue’s error distribution. Obviously, a
more traditional method for measuring equatorial coordinates by means of
wall circles was employed, and only then equatorial coordinates were recalcu-
lated into ecliptical ones on a special terrestrial globe.
Analysis of the distribution of the fractional parts in the catalogue coordi-
nates and error variances has shown that the longitudes were recalculated after
the measurement, which contradicts Ptolemy’s statement that the catalogue
§9
Astronomical and Mathematical Analysis of the Almagest
125
Figure 20(5). The real motion of the stars o 2 Eridanus, No. 1362, No. 1332,
No. 1290, No. 1363 (numeration according to D. Hofflit [323]) and the position
of the stars numbered in the Almagest as 778, 779, 780 (according to the
Almagest).
126
New Experimental and Statistical Methods
Chapter 3
y Ser
(real sky at the year f)
1900 AD *
100 AD x'
✓
*'
1900 AD Star No. 5913 in [323]
— * (real sky)
-O'""
Star 265 in the Almagest (»y Ser ?)
fl*50'- mean error in
the constellation Ser
1900 AD
Star No. 5913
✓''(real sky)
Figure 20(6). The real motion of the stars 7 Ser, No. 5913, No. 5909 (nu-
meration according to D. Hofflit [323]) and the position of the star No. 265
from the Almagest (according to the Almagest).
contains direct measurements (for a description of some other incoherences,
see [102]).
Second, attempting to date the catalogue via the motion of individual stars,
we discovered that for faint stars of magnitudes 3-6 which move appreciably, it
was necessary to analyze the identifications of the stars of today’s sky with the
Almagest's , since the traditional procedure already involved the data regarding
the stars’ own motion and the a priori dating the catalogue as belonging to the
start of the Christian era. As far as the bright stars which move appreciably
are concerned, the dates from a.d. 500 to 1800 are obtained. This enormous
variance and, therefore, the low accuracy of estimation, are related to the fact
that the rate of the stars’ own motion is too small compared to the errors in the
Almagest coordinates. Note that only five stars moving fast are included in the
Almagest. The author and V.V. Kalashnikov, G.V. Nosovsky, V.M. Zolotarev,
and I.S. Shiganov started investigating not with individual stars but with
the whole collection of the moving ones for the purpose of calculating the
date when they were observed on the basis of their totality. The results of
the statistical investigation were published in the paper by A.T. Fomenko,
V.V. Kalashnikov and G.V. Nosovsky, “When was Ptolemy’s star catalogue
in the Almagest compiled in reality? Statistical analysis”. Acta Applicandae
§9
Astronomical and Mathematical Analysis of the Almagest
127
Figure 20(7). The real motion of the star rj Cas and the position of the star
180 according to the Almagest.
Figure 21. The real motion of the star r Cet and the position of the star 723
according to the Almagest.
128
New Experimental and Statistical Methods
Chapter 3
Mathematicae 17: 203-229, 1989. This work was devoted to describing a new
statistical and geometrical procedure for dating ancient star catalogues by
considering numerical data contained in these catalogues and the known real
configurations of stars on the celestial sphere. The method was tested on
several star catalogues whose dates are well known (Tycho Brahe, etc.) and
on several artificially generated star catalogues. Then the same method was
applied to the Almagest The results obtained do not confirm the traditional
dating of the Almagest (2nd century A.D. or 2nd century B.C.) but shift its
dating to the Arabian epoch , i.e., to 600-1300 A.D.
The authors plan to publish the details of this research in a separate book.
Now, some concluding remarks about Halley’s discovery.
9.5. Halley’s discovery of the stars 9 proper motion and the Almagest
It is traditionally believed that the stars’ proper motion was first discovered
by Halley in 1718. Speaking of this fact, Kulikovsky says:
“Comparing the positions of Arcturus, Sirius and Aldeb&ran with their positions
in the Hipparchus catalogue, E. Halley (1656-1742) discovered their proper motion,
and that their ecliptical longitudes changed by 60 ; , 45', and 6 ; in the last 1850 years,
assuming that the catalogue was already dated as having been written in the 2nd
century B.C.” (namely, 1718 + 132 = 1850 years— A.F.) ([110], p. 219).
The first related question then is how he could have discovered Aldebaran’s
proper motion. It is because Aldebaran has shifted in the assumingly elapsed
time (allegedly, ca . 2,000 years) by 6', as we now know on the basis of modern
knowledge. However, the division value both in the Ptolemy and Hipparchus
catalogues was 10M It is useless to discuss the phenomenon if this value does
not exceed the accuracy of a measuring device, let alone the fact that the
actual accuracy of measurements was much worse than 10'. How then could
Halley have discovered Aldebaran’s proper motion if it had shifted by only 6'
in 2,000 years? Another question arises: What motion did Halley ascribe to
Arcturus and Sirius? Kulikovsky reports:
“In 1738, Cassini (1677-1756) precisely determined Arcturus’ proper motion, having
compared his measurements with Richet’s observations (?-1696) sixty years before”
([110], p. 219).
Therefore, Halley did not determine Arcturus’ proper motion “precisely” .
Nor could he have determined Sirius’ motion when it moved at a slower rate.
The natural and, to some degree, unexpected question arises: Did Halley,
at least in principle, discover the stars’ proper motion?
It will be appropriate to inform the reader that Halley was not at all the
first one to pose the question whether stars were moving, the problem hav-
ing been animatedly discussed in the 15— 16th century. Moreover, within the
framework of traditional chronology, this problem was investigated as early as
antiquity, or approximately 2,000 years before Halley. Tradition assumes that
this important problem was discussed by Ptolemy who came to the conclusion
that all the stars were stationary (which we know is wrong). Thus, posing the
problem cannot be ascribed to Halley.
§9
Astronomical and Mathematical Analysis of the Almagest
129
Why did astronomers not compare the positions of the stars in the sky to
the Almagest for the purpose of discovering their proper motion? As a matter
of fact, the very idea of such a computation, or comparison, also dates back
to Ptolemy and, therefore, was not at all new to a medieval astronomer. Such
attempts would have been natural and could have led to the conclusion that
the stars were moving: At least, the anomalous scattering in Ptolemy’s star
positions could have been taken as the stars’ proper motion. We have already
noted that the Almagesfs accuracy is not consistent with the catalogue divi-
sion value. Arcturus’ and Sirius’ motion could have been determined by the
early 17th-century astronomers (100 years before Halley), as well as by Halley
himself, on the basis of Tycho Brahe’s catalogue, its accuracy being V and
dating to 1582-1588 if we compare it with the Almagest.
Imagine that we are 16th-18th-century astronomers. Only the following two
points of view are possible.
(1) Assume that we are of Scaliger’s and Petavius’ opinion and associate the
second year of Emperor Antonine Pius’ rule, the date of the Almagest s obser-
vation time, with A.D. 138, in which case we should discover the stars’ proper
motion, resorting to the allegedly ancient, about 1,500-year-old catalogue, at
least proceeding from Arcturus, the brightest star in the sky. However, these
natural attempts were not fixed by the traditional history of astronomy in the
15— 16th century, though they would have to make us conjecture that Arcturus
was moving.
(2) Assume that we regard the Almagest as a comparatively recent doc-
ument, say, of the 10th-15th century, or as a source with unknown dating,
in which case the attitude toward the book would be quite different. If this
catalogue is assumed to be medieval, then the Almagesfs inaccuracy, which is
well known to the professional astronomers, and, moreover, the large division
value would simply not permit us to make such a recalculation (which could
not be made either if the catalogue date was unknown).
Since the history of astronomy reports nothing of the 16th- and 17th-century
astronomers’ attempts to discover the stars’ proper motion on the basis of the
Almagest , we are forced to conclude that they did not regard it as a sufficiently
ancient document.
Thus, a serious 16th- or 17th-century scientist should have made the con-
clusion that the accuracy of the coordinates in the Almagest did not permit
him to discover the stars’ proper motion if he regarded it as a medieval source.
On the other hand, if the Almagest had been regarded as an ancient docu-
ment, say, dating back to the 2nd century A.D., then the idea itself of using it
to discover the stars’ proper motion would have been very simple. It should
be remembered that the problem of the stars’ proper motion was important
in ancient times. It seems completely improbable for the idea to be new in
Halley’s time.
We shall now try to explain why the conclusion regarding the proper motion
of certain stars (e.g., of Arcturus or Sirius) could be made in Halley’s time,
though more or less precise values for their rate could not be found.
130
New Experimental and Statistical Methods
Chapter 3
The first precise star catalogue to the accuracy of V was made by Tycho
Brahe in 1582-1588. Arcturus’ and Sirius’ displacements in the 100 years
after Halley were about 3' and a little more than 2', respectively. With a
precise catalogue of the star positions as was available in the 18th century,
Arcturus’ and Sirius’ proper motion could already be suspected, though the
catalogue’s accuracy could not have as yet permitted one to determine the
rates of motion. It turns out that such a catalogue was indeed available in the
early 18th century. It was made by Flamsteed and was actually employed by
Halley even before its publication (with Halley having taken some incomplete
version from Isaac Newton, who was just investigating chronology).
Thus, in our opinion, Halley compared Flamsteed’s and Brahe’s catalogues,
and made the conclusion regarding the existence of Arcturus’, Sirius’, and
Aldebaran’s proper motion.
Meanwhile, Aldebaran’s motion indicated by Halley obtains the natural ex-
planation: He made use of the incomplete variants of Flamsteed’s catalogue in
which the Aldebaran information was given erroneously. Flamsteed regarded
his catalogue as not yet ready for publication at that time. It is known that
Halley made Aldebaran’s position more precise for the purpose (in his letter
to A. Sharp on September 13, 1718; see [109]).
Why then did Halley refer to Ptolemy’s Almagest as the basis for his com-
putations, and not to Tycho Brahe’s? Obviously, in Halley’s time, the tradi-
tional Almagest’s date, a.d. 138, “calculated” by Scaliger and Petavius had
already been canonized. To give his discovery more weight, Halley referred to
the Almagest , and not to Tycho Brahe, for the simple reason that the star
displacements discovered by him then looked more impressive. Calculating
Arcturus’ displacement from Brahe’s catalogue, Halley obtained only three
minutes, compared with the nominal accuracy of one minute in the former
catalogue, whereas, repeating the procedure for Ptolemy’s catalogue of (al-
legedly) A.D. 138, he naturally obtained a more substantial shift of about one
degree. It is apparent that Halley compared the shift with the Almagest’s nom-
inal accuracy of ten minutes, neglecting the problem of the actual accuracy
of the coordinates of the Almagest’s catalogue.
This reasoning makes us think once again that the Almagest had not yet
been regarded (in the 16th century) as an ancient, about 1,500-year-old doc-
ument and its “antiquity” was assumed known and canonized only early in
the 18th century.
Each of the above facts can also be separately “explained” within the tra-
ditional chronology framework, but, taken all together, they most probably
indicate that the Almagest was written between the 10th and the 16th century.
Still, in contrast to Morozov, I do not at all believe that the Almagest was
fabricated. More than that, in my opinion, at any rate, its first edition is an
original created in the 15th— 16th century for immediate scientific purposes
(see the substantiation below).
§10
Archaeological Dating Methods
131
§10. Archaeological Dating Methods
10.1. Classical excavation methods
Certainly, the reader may ask how things are with the other methods of dating
classical historical sources and monuments. Modern archaeologists painfully
speak of “ignorant diggers” of the earlier centuries, who, hunting for valuables,
mercilessly disfigured numerous monuments.
“When the artifacts were delivered to the Rumyantsev Museum (excavations of
1851-1854 — A.F.), they were, in the true sense of the word, a shapeless heap, without
any indication from which excavated hill they came . . . The grandiose excavations of
1851-1854 . . . will be long deplored by science” ([65], 12-13).
At present, excavation methods have become more sophisticated; unfortu-
nately, it is quite rarely that they apply to excavations of ancient historical
spots, all of them having already been “touched” by earlier “diggers”.
Here are, in a nutshell, the archaeological dating basics. Greek vessels of the
Mycenaean culture were discovered in the tombs hailing from the Egyptian
18th and 19th dynasties. The dynasties and culture were, therefore, regarded
by the archaeologists as having been simultaneous events. The same vessels,
or “similar ones”, were found along with clasps of special form in Mycenae,
and similar pins in Germany, near burial urns. A similar urn was found in
Germany, and in this urn a pin of a new form. A similar pin was then also
found in Sweden, in the so-called hill of King Bjorn. Thus, this hill was dated
to the 18th and 19th dynasties of Egypt [66]. It was meanwhile discovered
that the Bjorn hill
“. . . could not belong to Bjorn, king of the Vikings, and was made well two thousand
years before” ([66], pp. 55-56).
It is not clear what should be understood by the “similarity” of the findings;
hence, all these, and similar, methods are based on unlimited subjectivism and
traditional chronology. The newly-found artifacts are compared with “simi-
lar” finds dated earlier on the traditional basis. A change in this “scale” will
automatically alter the chronology of recent archaeological findings.
The excavations at Pompeii can be a good illustration of the problems
which arise in dating archaeological material. The medieval author of the 15th-
century Sannazaro Jacopo wrote that, as he approached Pompeii, its towers,
buildings, theatres, and temples became visible, not touched by centuries!?
([47], p. 31). But it is impossible from the traditional point of view, because
Pompeii is regarded to have been buried by the volcanic eruption of a.d. 79.
Traditional history insists that Pompeii’s ruins were completely covered by
sand for many centuries till the 18th century. Hence, the modern archaeologists
are forced to treat this citation from Sannazaro thus:
”... in the 15th century, certain of the Pompeii buildings became visible in a layer
of sand ...” ([47], p. 31).
Therefore, it is assumed that Pompeii was again (after the 15th century)
132 New Experimental and Statistical Methods Chapter 3
“covered by sand”, since its remains were discovered only in 1748 and Hercu-
laneum’s in 1711 ([47], pp. 31-32). The excavations were barbaric.
It is now hard to determine the extent of damages caused by the vandalism
of the time. If a picture seemed not to be very pretty, then it was broken into
pieces, thrown away as waste. Separate letters were taken off the bronze in-
scriptions of marble tables. Souvenirs for tourists were fabricated of sculptural
fragments, sometimes even with the images of saints, certain of the “fabrica-
tions” possibly being originals [68].
There were obviously medieval pictures on Pompeii’s walls, such as a hang-
man under a hood, a warrior with a cuirass, helmet and visor (!), and so
on ([47], pp. 210-211). Surgical instruments indistinguishable from medieval
ones, and of extremely high quality, were found. An emperor with the double
name Valens-Nero (translation due to Morozov) was mentioned on one of the
official inscriptions, though, according to traditional history, these are two
different men separated by ca. 300 years [47].
The 20th-century archaeologists and historians paid much attention to a
particular strange process. Most of the ancient monuments in the last 200-
300 years, starting when continuous observations began for some reason or
other, were destroyed much more than during the previous centuries or even
millennia. In particular, they include the Parthenon, Colosseum, theatre of
Epidaurus, Venetian buildings, Indian temples, and others. “Modern” indus-
try is usually to blame, but nobody investigated thoroughly the influence of
“modern” civilization on stone buildings. It is natural to suggest that they
were not as ancient as asserted traditionally, and were being destroyed at a
natural rate.
10.2. Numismatics
“Numismatics as a science was formed comparatively late. The intermediate stage
from simple collections to scientific research methods . . . was presumably the very end
of the 18th century” ( [l 00] , pp. 13-14).
The numismatics scale and dating and the relative chronological ordering
of ancient coins on the time axis are wholly based on traditional chronology
regarded a priori as known, and they were established earlier on the basis of
written evidence.
“Material was gathered from almost 200 years of numismatics in the collectors’
hands, but no attempts to study it critically were made” ([100], pp. 13-14).
Most ancient coins were supplied with abbreviated formulas, or anagrams
with ambiguous dating, and without detailed inscriptions. The dates on coins
were sometimes based on calendars whose relation to the counting of the mod-
ern year is still suitable to a special study. It is assumed that, from the 8th until
the mid- 13th century, the medieval gold coins of Rome practically vanished
in Italy. However, there were many much more ancient Roman gold coins.
Our conjecture is as follows: these ancient coins are in reality the medieval
§10
Archaeological Dating Methods
133
gold coins, and minting them in medieval Rome possibly had not stopped.
But the latest chronologists have shifted these medieval coins to the past.
10.3. The dendro chronological method
Certain physical procedures are applied to the dating, for example, of den-
drochronology or the radiocarbon method. However, dendrochronological
scales were established to account only for several hundred years backwards,
which does not permit us to date ancient European buildings.
“Many European scientists employed dendrochronology for dating — But, it soon
became clear that the matter was not so simple. Trees in European forests are no more
than 300-400 years old The leaf-bearing wood is hard to analyze. Its blurred rings
very reluctantly speak of past times Reliable archaeological data turn out to be
unexpectedly insufficient” ([69], p. 103).
American dendrochronology based on the Douglas fir, bristlecone, and yel-
low pine is much better off, but deals with an area far away from the “areas of
antiquity” ([69], p. 103). Moreover, there are always many completely unac-
countable factors, such as local climatic conditions, soil composition, humidity
variations, landscape, and others, which alter the graphs of the annual ring
thickness essentially ([69], pp. 100-101). It is important that the construction
of dendrochronological scales was carried out on the basis of already existing
traditional chronology ([69], p. 105); therefore, any change in the chronology
of documents will alter the dendrochronological scales.
The methods of dating by determining column deformation, river sediment
accumulation rate, stone building surface layer weathering rate, sound prop-
agation rate in bones, and others, remain extremely inaccurate and depend
on traditional chronology assumed a priori as known, with the principal dif-
ficulty being the absence of a reliable absolute scale and the impossibility to
“calibrate a method” for dating ancient artifacts.
10.4. The radiocarbon method
The most popular is the radiocarbon method claiming the independent abso-
lute dating of antique monuments. However, as radiocarbon dates were accu-
mulated, most serious difficulties arose; in particular,
“. . . one had to think of another problem. The radiation piercing the atmosphere
varies for many space reasons, the quantity of radiocarbon therefore changing time.
It is then necessary to find a technique which would permit us to take them into
account. Besides, . . . huge amounts of carbon from burning charcoal, anthracite, oil,
pect, shale and their by-products are continuously thrown into the atmosphere. What
influence does this have on the rise of the radioactive isotope quantity? To achieve
a method for determining the true age exactly, one has to compute the complicated
corrections reflecting the changes in the composition of the atmosphere during the last
millennium. Along with certain difficulties of technological nature, these doubts led
to uncertainty regarding the accuracy of a lot of datings achieved by the radiocarbon
method” ([69], p. 72).
134 New Experimental and Statistical Methods Chapter 3
Its inventor W. Libby has been absolutely sure of the correctness of tradi-
tional historical datings:
“Throughout Roman and Egyptian history, we have no disagreements. We haven't
had very many measurements to make (! — A.F.) because in general the archaeologists
know the dates better than we can measure them, and it is usually as a favour that
they give us samples (which are by the way, binned in the measuring process — A.F.)”
([70], p. 24).
This confession is significant, because traditional chronology revealed its
difficulties particularly for the regions and epochs with only a few datings by
the method. As to the small number of test measurements for antique artifacts,
which, nevertheless, were taken, for example, in dating H. Breasted’s collection
(Egypt),
. . our third object from Egypt turned out to be modem! It was one of the . . .
collections, supposed to be from the fifth dynasty (i.e., 2563-2423 B.C. according to
[7], about 4,000 years ago — A.F.). This was a dark day” ([70], p. 24).
To “save” traditional chronology, the object was declared a fake ([70], p. 24),
since no one had doubted the traditional chronology of ancient Egypt.
“In support of the basic assumption, they (believers in the method — A.F.) list a
series of indirect proofs, arguments, and computations whose accuracy is not high,
treatment ambiguous, and the main proof given by test radiocarbon determination
of samples with an a priori known age.. . . However, as soon as it comes to the dis-
cussion of the test results in dating the historical artifacts, everyone nods to the first
experiments, a small (! — A.F.) series of samples ...” ([66], p. 104).
The absence (also recognized by Libby) of any representative test statistics
for the objects whose reliable dating is known a priori and a divergence of
many thousand years long make us question the appropriateness of applying
the method to the time interval under consideration. It does not mean, how-
ever, that it cannot be applied for the purposes of geology, where errors of
several thousand years do not matter.
Libby writes:
“There has been no serious shortage of materials back to about 3,700 years with
which we could check the accuracy and reliability of dating by carbon- 14 ... I would
say, from talking with historians, that they would stake their lives on 3,750 years, but
with anything older than that, they begin to shake a little bit” ([70], p. 24).
In other words, as the inventor himself recognizes, the radiocarbon method
was widely applied in particular where the obtained results were hard or
impossible to check against other written evidence.
“Some archaeologists, while not doubting the scientific principles of carbon-14 dat-
ing, suggested that the method still involves a considerable margin of error due to
factors which are not yet understood” ([70], p. 29).
Could it perhaps be that these errors still did not present obstacles to at
least rough dating? As it turns out, the situation is more serious: The errors
are too large and chaotic and attain a span of 1,000-2,000 years on dating
objects that are 1-3-millennia old.
§10
Archaeological Dating Methods
135
The radiocarbon dates led to
. . confusion among the archaeologists. Some have accepted the physicists’ in-
dications . . . with characteristic admiration.. . . They hurried up to reconsider the
chronological schemes (which, therefore, were not so firmly established? — A.F.) ...
The first who declared himself openly against the radiocarbon method was Vladimir
Milojdic . . . not only storming at its practical applications, but also . . . strongly criti-
cizing the theoretical foundations. . . . Comparing individual measurements of modem
samples with the average figure, he based his skepticism on a series of brilliant para r
doxes. The shell of a living American mollusk with radioactivity level 13.8 turns out
to be rather old, being about 1,200 years of age if we compare it with the average
absolute norm, 15.3. Its radioactivity being 14.7, a blooming wild rose from North
Africa will be ‘dead’ for as long as 360 years . . . whereas an Australian eucalyptus of
radioactivity 16.31 ‘does not exist* for the physicists at all, and will come into exis-
tence in 600 years. A Florida mollusk with 17.4 radioactivity units will be ‘born’ only
in 1,080 years. ... In the past, radioactivity level was no more uniform than today,
and similar variations are also possible for ancient artifacts. Here are some examples:
The radiocarbon dating of a sample from a medieval altar in Heidelberg . . . showed
that a tree employed to repair it had not grown at all! ... In a cave in Iran, the lower
levels were dated to 6054 i 415 and 6595 ± 500 years B.C., while the upper to 8610
± 610 B.C. Thus . . . they obviously were formed in reverse order, and the upper one
turns out to be 2,556 years older than the lower. There are many similar examples to
follows. ... V. Milojcic calls on the physicists and their ‘contractors’, at least, to stop
correcting ‘critically’ the radiocarbon dating results, and cancel the ‘critical’ censor-
ship when the data are published. He asks the scientists not to reject the dates which
seem improbable for some reason or other, and publish all the results without excep-
tion. He talks the archaeologists into ceasing the practice of acquainting the physicists
with an approximate age of a find before its radioactive analysis, with holding the in-
formation until the figures are published! Otherwise, it will be impossible to establish
how many radioactive dates coincide with the reliable historical ones, or determine
how reliable the method is. Besides, on ‘editing’, the subjective attitude influences how
the obtained chronological scheme looks. B.g., in Groningen, where the archaeologist
Becker, a long-time supporter of the shorter chronology (of Europe — A.F.), is working,
the radiocarbon results are also obtained to be low for ‘some unknown reason’, but
Schleswig and Heidelberg, where Schwabdissen and others have long been inclined to
the longer chronology, the radioactive dating of similar materials leads to much earlier
dates” ([66], pp. 94-95).
In my opinion, any comment is superfluous.
In summary,
(1) Theoretical accuracy of carbon- 14 dating, which can be achieved for
comparatively “young” objects, equals about 1,000-1,500 years; hence, un-
fortunately, its application to dating antique artifacts seems to be doubtful.
We should first investigate for as large a set of reliable data as possible the
dependence of radiocarbon dates on the variation of carbon-14 content, in-
tensity of cosmic radiation, distance to large water reservoirs and volcanoes,
enormous quantities of various substances in the atmosphere, and so forth,
with all this, in addition to other factors, essentially influencing the dating,
as we have demonstrated above.
(2) Other physical methods are sufficiently less accurate and have been
carried out, basically, for the purposes of geology; today, their application to
dating antique artifacts is unreal.
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New Experimental and Statistical Methods
Chapter 3
Lastly, great successes in working out the radiothermoluminescent method
permitting us to date ceramic artifacts and burned clay have been made.
However, here, too, numerous effects influence the dating of truly old artifacts,
which are now barely taken into account. In particular,
“. . . one should know the quantity of radioactive elements not only in the substance
of which an object is made, day, but also in the environment surrounding it. If a piece
of ceramics had been buried for centuries, then the quantity of uranium, thorium and
potassium in the soil or rock should be known” ([92], p. 113).
In 1972, the physicist S. Fleming made the method substantially more so-
phisticated in order to eliminate the dependence of the dating on radiation
in the environment, which, certainly, almost always remains unknown and is
not subject to estimation. Nevertheless,
“. . . dating authentic terracotta of the Renaissance by the method of thermolumi-
nescent analysis is related to certain difficulties” ([92], p. 138).
“Since the exhibits kept in variable environment were dealt with, the dose of exter-
nal radiation was not simple to find” ([92], p. 139).
Still, the dating was carried out, but only because the researchers did pos-
sess authentic samples dated reliably as belonging to the Renaissance (from
written sources). The samples dated questionably were compared just with
these. It is obvious that the method cannot be applied to older, say, “antique”
artifacts, without any independent and trust-worthy written information for
dating them. Nor do we speak of a practically unrealizable opportunity to
record the variable environment in which antique ceramics are kept in the
museums (see above).
§11. Astronomical Dating. Ancient Eclipses and Horoscopes
At present, special tables, or astronomical canons, are available which list
eclipse dates, umbra ranges, astronomical phases, and so forth, on the basis
of the lunar theory. If an eclipse was given a sufficiently detailed description
in antiquity, then the list of its observed (descriptive) characteristics can be
compiled. Comparing them with the tables, we can attempt to find a suitable
eclipse in the canon. We shall date the eclipse in question if we succeed. It
may turn out that the textual description is satisfied not only by one, but
already several eclipses from the astronomical canon, in which case the dating
is not unique.
While taking up certain problems of celestial mechanics, I noticed a possible
relation of this familiar jump in D n to the results of Morozov, connected with
the dating of ancient eclipses. My investigation of this problem, and a new
calculation of D", have shown unexpectedly that the new curve obtained
for D N is of another qualitative character; in particular, the enigmatic jump
vanishes completely, while D n oscillates about one and the same constant
value coinciding with the modern one (see [3]). In short, this result is reduced
to the following.
§11
Astronomical Dating . Ancient Eclipses and Horoscopes
137
The earlier computations of D n were based on the dates of ancient eclipses
according to traditional chronology. All attempts to account for the strange
jump in D" did not touch upon the problem whether or not the dates of
eclipses regarded today as antique and early medieval were determined cor-
rectly. In other words, how well do the parameters of an eclipse described
in a document, and the computed characteristics of that authentic eclipse al-
legedly described in the text to be dated, correspond to each other? A method
of independent dating was suggested by Morozov in [1], namely, that all pos-
sible eclipse characteristics are extracted from a text under investigation, and
then the dates of all the eclipses with these characteristics are mechanically
extracted from the astronomical tables (canons). He discovered that, under
the pressure of the established traditional chronology, the astronomers did
not consider the whole spectrum of dates obtained: they took only those dates
which fit in the time interval a priori dictated by historical tradition. It turned
out that this practice often led the astronomers to the impossibility of discov-
ering in the required century an eclipse precisely answering to the description
in the document, while being forced, in most cases, and still not questioning
the whole system of chronology, to resort to doubtful solutions, for example,
to indicate an eclipse no more than partly satisfying the description.
Revising the dating of supposedly ancient eclipses, Morozov discovered that
the information about them could be separated into two categories:
(1) Short and vague reports without particulars, with it often being unclear
whether an eclipse at all is described; in this category, the astronomical dating
is either senseless or results in so many possible solutions that they can fall
into practically any historical epoch.
(2) Lengthy and detailed reports, with often unambiguous astronomical
solutions that admit only two or three versions. All the eclipses in this category
turn out to have received nontraditional dating, which did not assign them
to the traditional interval from 1000 B.C. to A.D. 400, but considerably later
(sometimes, many centuries), i.e., to A.D. 500-1600. Nevertheless, Morozov
did not analyze the latter, assuming that traditional chronology from 300 to
1800 was basically “true”, and that no contradiction would surface.
Proceeding with the research initiated [1], I also subjected the eclipses from
400 to 1600 to an analysis. It turned out that the effect discovered in [1] for
ancient eclipses could consequently be extended to those normally dated to
A.D. 400-900, which means that two situations are realized: 1) there are many
equivalent astronomical solutions (here astronomical dating is senseless); 2)
there are only one or two solutions; in this case they fall into the years 900-
1700. It is only since about A.D. 900, and not 400 as supposed [1], that the
consistency becomes satisfactory for the traditional eclipse dates given in the
canon [73] with Morozov’s results and is reliable only since A.D. 1300. Thus,
we obtain the shift of the dates of many “ancient” eclipses in the medieval
epoch from A.D. 900 to 1600.
An analogous “forward date shift” was also discovered by Morozov for the
so-called horoscopes [1]. Five planets are seen with the naked eye; moving
138
New Experimental and Statistical Methods
Chapter 3
along the ecliptic, they describe approximately the same circular path in the
sky, called zodiac and divided into 12 sections. Astrology enjoyed enormous
popularity in ancient times. A horoscope is the position of the planets with
respect to the signs (constellations) of the zodiac. Fixing it, and knowing the
sidereal periods at a particular point in time, we can calculate the planets’ po-
sitions in the past or future by marking off their integral multiples backwards
or forwards. However, the realization of this simple idea implies cumbersome
calculations. Similar to the eclipse canons, there are special tables making dat-
ing of their ancient descriptions possible (we should not confuse them with
those in the contemporary sense of the term). If the planets’ position is given
some textual description, then, as in the case of the eclipses, we can obtain
the dates of all horoscopes with suitable characteristics from the tables.
Again similarly to the eclipses, it turns out that under the pressure of the
traditional chronology already established, and unable to find a suitable horo-
scope “in the required epoch”, the astronomers forcibly resorted to doubtful
conclusions and deviations from the text. Morozov analyzed the most famous
ancient horoscopes in [1] and discovered that all those with detailed descrip-
tions (or graphic representations) obtained medieval or even late medieval
datings if we employed independent methods for them.
We illustrate with some typical examples. Numerous attempts by P. Laplace,
J. Fourier, A. Letronne, J. Biot, and K. Helm to find a suitable solution to
a horoscope represented on the “round” and “long” zodiacs of the Dandarah
temple in Egypt were not crowned with success ([1], Vol. 6, pp. 664-665,
Figs. 672-673, Figs. 133, 135). The temple and its horoscopes are now dated
to 30 B.c. and a.d. 14-37. Nevertheless, two exact astronomical solutions do
exist. The first solution, A.D. 568 and 540, was discovered by Morozov [1]; the
second solution, A.D. 1422 and 1394, was discovered recently by the Moscow
physicists D.V. Denisenko and N.S. Kellin (for details, see Vol. 2 of the present
book).
In 1857, H. Brugsch discovered an ancient Egyptian sarcophagus whose
inner cover represented the starry sky with a horoscope ([1], p. 696, Fig. 139).
The whole burying ritual, ancient demotic writing, and so forth, dated the find
to not earlier than the 1st century a.d. The attempts by the astronomers to
date the horoscope as having originated at around the turn of the millennium
failed. However, not only does the exact solution exist again, but it is unique
in the whole interval of history: namely, A.D. 1682!
In 1901, Flinders Petrie discovered a cave in Upper Egypt with an ancient
Egyptian tomb and two horoscopes with the dates of the deaths of a man and
his son buried there [1]. In the whole historic interval, there exists a unique
solution ideally satisfying all the conditions of the problem, namely, A.D. 1049
(the father’s horoscope) and 1065 (the son’s horoscope). The son died 16 years
after the father.
The dating thus described also explains an exceptionally fine state of these
ancient Egyptian pictures made in watercolour. The above-mentioned work
also dates the horoscopes described, for example, in ancient biblical texts. We
§11
Astronomical Dating . Ancient Eclipses and Horoscopes
139
can compile a vocabulary of the terms and standard phrases frequently oc-
curring in the preserved medieval astronomical literature and used to denote
planets, constellations, and so on. Then, when encountering an ancient verbal
description involving any of these terms, an attempt can be made to date it,
and spell it out, as a horoscope by means of this vocabulary. Apparently, the
first author who mentioned that the Revelation of John contained the verbal
description of a horoscope was E. Renan [74]. He was not an astronomer, how-
ever, and did not date it, though the solution of the question is of considerable
interest due to the existing problem of dating the Revelation of John.
J. Sunderland declares that to take the end of the 1st century A.D. or even
whatever other time as the date of the creation of the Revelation of John
is to face great difficulties ([49], p. 135). However, it turns out that, though
not unique, the exact solution does exist, too, namely, A.D. 395 and 1249 [1].
The date A.D. 395 differs from the traditional one for the Revelation of John
by 300 years, whereas the second, A.D. 1249, differs already by 1,100-1,150
years. Accordingly, we stress that the bulk of Morozov’s astronomical datings
should be reconsidered. As a matter of fact, the assumption that traditional
chronology since a.d. 300 is correct has often made him discontinue the com-
putations and dwell, in most cases, on the first suitable solution discovered
without turning to the late Middle Ages, which is vividly justified by the ex-
ample of the Revelation of John. The second solution, A.D. 1249, was rejected
by Morozov as “too late”.
He wrote: “Hardly anyone will say that the Revelation of John was written
on September 14, 1249 ...” ([1], Vol. 1, p. 53).
Our point of view is that we should investigate the whole spectrum of
possible solutions, the more so because there are sound reasons to believe (see
below) that traditional chronology until the end of the 13th century should
be completely revised.
Another example is given by the dating of the famous eclipse accompanying
the crucifixion according to the early Christian authors such as Synkellos,
Phlegon, Aphricanus, Eusebius, and others. The traditional date of April 3,
A.D. 33, cannot withstand even minimal criticism [1]. In spite of considerable
controversy surrounding the characteristics of this eclipse, which is repeatedly
discussed in the literature (e.g., [73]), we can try to date it. The first exact
solutions (found by Morozov in [1]) turn out to fall between 200 B.c. and
A.D. 800, namely, A.D. 368. The calculations in [1] were not extended to the
later centuries for the above reasons. I did extend the computations to the
whole interval of history up to a.d. 1600 and unexpectedly discovered another
precise solution, i.e., April 3, a.d. 1075, which differs from the traditional one
by one millennium, and by 700 years from that found in [1]. From the formal
standpoint, the two solutions are equivalent, and we have to base ourselves on
another argument in order to make a final choice. We recall once again that
the agreement of the traditional astronomical dates with the calculated ones
becomes satisfactory only beginning with the 10th century, and reliable only
since the 13th century.
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New Experimental and Statistical Methods
Chapter 3
The horoscopes of the Old Testament turn out to be medieval, which is
in startling contradiction with the traditional point of view referring to the
events described in the Old Testament as having occurred hundreds of years
before Christ [1].
§12. New Experimental and Statistical Methods of Dating Ancient
Events
12.1. Introduction
In my opinion, the main goal is the creation of novel and statistically inde-
pendent methods of dating ancient events. It is only subsequently that we can
turn to the analysis of the whole of chronology on the basis of the obtained
results. One method, even as effective as the astronomical one mentioned
above, is absolutely insufficient for a profound analysis of the dating problem
due to its exceptional difficulty and requires a cross verification of the dates
by different techniques.
I attained this goal as follows.
(1) New experimental and statistical procedures for dating ancient events
were worked out [2], [3], [4], and [5].
(2) Their efficiency was checked experimentally against a sufficiently large
amount of information regarding medieval history, confirming the cor-
rectness of the obtained results.
(3) The methods were then applied to the ancient historical chronological
data, which resulted in important laws in ancient and medieval chronol-
ogy and history [2]— [5] .
(4) The laws were collected and systematized as the global chronological
diagram (GCD), which I briefly described in [2]-[5].
(5) The hypothetical mechanism of the origin of versions of the traditional
chronological ancient and medieval history could be constructed from
the GCD.
We shall now give a brief account of some of these methods.
12.2. Volume graphs for historical chronicles. The maximum
correlation principle. Computational experiments and
typical examples
Suppose that a certain period between the two years A and B of the history
of a state is described annually in some sufficiently long text X by means
of chronicles, annals, and so forth, which is broken, or could be broken, into
pieces, “chapters” X(t ), each of which describes one year t. We can count
their volume, for example, the number of words, signs or pages, and represent
the obtained values graphically by marking the years t off on the horizontal,
and the chapter volumes on the vertical scale (see Fig. 22).
In general, the corresponding graph will have a different form for another
text Y, since the authors’ propensities influence the volume distribution. For
example, an art history and a military chronicle will place accents differently;
§12 New Experimental and Statistical Methods of Dating Ancient Events 141
Figure 22. Maximum correlation principle for the volume functions of depen-
dent historical chronicles.
the amount of information with respect to different years will be distributed
differently, too. How essential are these differences, i.e., do there exist charac-
teristics of the volume graphs which are only determined by the time interval
(A, B) and the country T, and which uniquely characterize all (or almost all)
texts describing them? The important characteristic of the volume graph turns
out to be given by the years in which the graph shows peaks, i.e., attains local
maxima. They indicate the years described in more detail. In general, differ-
ent years will receive detailed descriptions in different, independent chronicles.
Let C(t) be the volume of all texts describing the events of a particular year t
and having been written by contemporaries. According to our informat ion- loss
model, there will be more information left from those years in which especially
many texts were created . It is difficult to verify the hypothesis in its present
form, because the graph of C(t) is unknown to us, and the texts and their
information are gradually lost. However, we can check one of the corollaries,
namely, later authors X and Y who were not contemporaries to those ancient
events and who described the same period, had to make use of approximately
the same collection of the preserved texts; therefore, they would “on the av-
erage” speak more of those years from which more texts survived, and less of
those from which less was known. In other words, the authors “on the aver-
age” must have increased the number of particulars in writing about the time
from which more texts were available. The maximum correlation principle can
eventually be formulated as follows:
The volume graphs for the chapters of two dependent texts X and Y which
describe the same period (A, B) and the state T must attain local maxima , or
form peaks , simultaneously , i.e., years described in X andY in detail should be
close or coincide. Conversely , if two texts X andY are known as undoubtedly
independent and describe either different periods (A, B ) and (£7, D) of the
same length or different states, then their volume graphs reach local maxima
at different points if we let ( A,B ) and (C, D) coincide .
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New Experimental and Statistical Methods
Chapter 3
The principle will be justified if for most pairs of authentic and sufficiently
large dependent texts depicting the same events, their volume graphs form
peaks at about the same time. Meanwhile, the peaks themselves may be much
different, and there must be no correlation between the peaks of the graphs
for authentic independent texts. The simultaneity of the peaks of the volume
graph should certainly be only approximate for concrete dependent texts.
To estimate their nearness quantitatively, we shall calculate the square sum
<p(X,Y) of the distances <pk in years from the point of a peak indexed by
k of the volume graph for X to the kth peak of the volume graph for Y.
If both graphs have peaks simultaneously, i.e., if the moments of the peaks
with the same indices coincide, then all (fk are equal to zero. Considering a
sufficiently large fixed stock of different authentic texts H , and calculating
<p(X,H) for each of them, we then only select H with <p(X } H) less than
or equal to <p(X,Y). Finding their part in the entire stock of JZ, we shall
obtain a coefficient which (under the hypothesis regarding the distribution of
the random vector H) can be interpreted as the probability d(X,Y). If the
coefficient is “small”, the texts X and Y are dependent; if it is large, they are
independent, i.e., they describe different events.
In 1978-1980, I performed a vast computational experiment in order to
evaluate d(X,Y) for several hundred pairs of concrete historical texts, includ-
ing chronicles, annals, and so forth (see the details in [2]). It turned out that
d(X,Y) very clearly distinguished between dependent and independent tex-
tual pairs. I discovered that for all authentic texts (X, Y) under investigation
and describing different events known a priori as such (or different historical
epochs or states), i.e., for independent texts, d(X, Y) varied from 1 to 1/100
(for ten to fifteen local maxima); conversely, if two texts X and Y were known
a priori as dependent and described the same events, then d(X,Y) did not
exceed 10“ 8 (for the same number of maxima). A typical example is shown in
Fig. 22. Namely, let X be Sergeyev’s monograph Essays on the History of An-
cient Rome , and Y be Livy’s History of Rome. Then d(X, Y) is 210” 12 , which
implies the interdependence of the texts which are both describing the same
period in Roman history. If, however, we take the first part of X as A', and
the second half as Y', i.e., certainly independent texts, then p(X', Y') = 1/3.
We now give another example of undoubtedly dependent texts, namely, X
is the Nikiforovskaya chronicle and Y is the SuprasVskaya chronicle [19]. Both
volume graphs for the time period from 850 to 1255 exhibit peaks practically
simultaneously (Fig. 23) in the same years, and d(X, Y) = 10~ 24 .
The experiment compared ancient texts with ancient ones, ancient with
modern ones, and modern with modern ones.
Along with the chapter volume graphs, other quantitative textual character-
istics were investigated, namely, the quantity graphs of the mentioned names,
given the number of textual mentions of individual years, other fixed text
reference frequency graphs, and so forth [2]. The same maximum correlation
principle turns out to be valid for all these characteristics: The graphs of un-
doubtedly dependent texts form peaks almost simultaneously, and the peaks are
§12 New Experimental and Statistical Methods of Dating Ancient Events 143
Figure 23. Volume functions of two dependent historical chronicles: the Niki -
forovskaya chronicle and the Suprasl’skaya chronicle ( Russian medieval his -
tory).
not correlated on the graphs of undoubtedly independent texts , which permits
us to suggest a new dating method for ancient events.
Let Y be a historical text describing unknown events whose absolute dating
has been lost. Let the years t be counted since an undated event of local
importance, for example, the foundation of some city or the crowning of a
king. We construct the volume graph for the chapters of Y and compare it
with those of other dated texts. If among these texts, a text X with small
d(X y Y ) is discovered which is of the same order as for pairs of dependent texts,
i.e., if it does not exceed 10~ 8 , then we can conclude with sufficiently large
probability, becoming greater as d(X,Y) decreases, that the events described
coincide.
The method has been verified experimentally for a priori dated medieval
texts. The obtained dates coincided.
We illustrate this with the example where Y is the Dvinsky chronicle (the
shorter version) describing the events in a 327-year-long interval [19]. Going
through the list of chronicles in the Complete Collection of Russian Chronicles,
we discover a text X whose volume graph forms peaks almost in the same
years as that for Y (after the time intervals {Ay B) and (C, D) have been
made to coincide), where d(XyY) = 2 • 10~ 25 . It turns out that the text X
is the complete version of the same Dvinskaya chronicle [19], with {A y B) =
1390-1717. The obtained dating of the text Y coincided with its standard
dating.
12.3. Method of recognition and dating the dynasties of ancient
rulers. The small-distortion principle
Suppose a historical text has been found, describing an unknown dynasty of
rulers with known rule durations. The question arises whether this dynasty
is, in fact, new and unknown (hence, requiring a dating), or whether it is one
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New Experimental and Statistical Methods
Chapter 3
of the known dynasties described in unusual terms, with the names of rulers
altered, and so forth. The answer is given by the method described below
(see its short description in [4] and [5]). Consider a sequence of authentic
rulers in a state. We call this sequence an authentic dynasty . Its members
must not necessarily be relatives. The same authentic dynasty can often be
described in different documents (by different chroniclers) and from different
standpoints estimating the rule lengths differently, and so on. But there do ex-
ist “invariant” facts whose descriptions depend less on the propensities of the
chroniclers, for example, rule durations. Normally, there are no special reasons
for which a chronicler would considerably or intentionally distort this number.
Nevertheless, chroniclers often face difficulties in calculating the duration of a
king’s rule. Naturally, for example, due to the incompleteness of information,
distortions in the documents, and so on, they have sometimes caused different
chroniclers to supply different figures which were, in their opinion, the rule of
the same king. Such discrepancy can be characteristic, e.g., of the pharaohs in
Brugsch’s tables and [6]. Thus, describing an authentic dynasty, each chron-
icler calculates the durations of the kings’ rules in his own way and obtains
a sequence of numbers (A i) A 2 y . . , A*), with A p representing (possibly, with
an error) the authentic rule of a king indexed by p, and with k being the total
number of kings in the dynasty. This sequence (extracted from the chroni-
cle) will be called a numerical dynasty . While describing the same authentic
dynasty, another chronicler will possibly ascribe other rule durations to the
same kings and obtain another numerical dynasty (Bi, f?2, • • • , B* ). Thus, the
same authentic dynasty described in different chronicles can be represented as
different numerical dynasties. We now formulate the “small-distortion princi-
ple”:
If two numerical dynasties are “little” different from one another, then they
represent the same authentic dynasty and are two variants of its description
(in which case the numerical dynasties will be said to be dependent ); if, how-
ever, two numerical dynasties represent two different authentic dynasties, then
they are “considerably” different from each other (in which case we call them
independent ).
In other words, chroniclers “little” distort authentic dynasties in writing a
chronicle; at any rate, the discrepancy which can arise is less than that be-
tween different authentic dynasties. The above assertion should be verified.
In case it is valid, we discover an important, and not at all obvious, prop-
erty characterizing practically all chroniclers of antiquity, namely, numerical
dynasties that arise from describing one authentic dynasty are different from
each other — and also from the prototype — by less than two different authentic
dynasties.
It turns out that to estimate the “nearness” of two dynasties, we can in-
troduce a numerical coefficient A (M, B) similar to d{X y Y), which still has
the same meaning as probability. We describe it without going into details. It
is convenient to represent a numerical dynasty as a graph by indicating the
numbers of kings on the horizontal, and their rule duration on the vertical
§12 New Experimental and Statistical Methods of Dating Ancient Events 145
scale. We will say that a dynasty II is “similar” to the dynasties M and H
if its graph differs from that of M by not less than the graph of H differs
from that of M (see the details in [2], [4], [5]). We take as A (M, if) the part
made up by the dynasties that are “similar” to M and H in the set of all
dynasties, i.e., the ratio of the number of all dynasties “similar” to M and
H to the total number of dynasties fixed in the chronicles. Rule durations
may be determined wrongly by the chroniclers, and we actually extract only
approximate values. The probability mechanisms leading to the errors can be
described mathematically. Besides, we have then taken into account another
two of the chroniclers’ possible error, namely, mixing up neighbouring kings
and their replacement by one whose rule duration equals the sum of theirs.
The small-distortion principle should be verified. In 1977-1979, I investi-
gated J. Blair’s tables [6] containing all the basic chronological data pertaining
to the history of Europe, the Mediterranean, the Near East, and Egypt from
4000 B.c. to A.D. 1800. The data were checked and supplemented with the
information from 14 more modern tables. For all the epochs in the history of
the regions, a complete list of all 15-term dynasties was made, namely, of all
groups consisting of 15 consecutive kings. Each could fall in several different
15- term dynasties, i.e., dynasties could “overlap”.
We can compute A (M, H) or any two 15-term dynasties M and H . My com-
putational experiment then showed that the small-distortion principle could
be regarded as fully reliable, namely, for two a priori dependent dynasties,
A (M, H ) is always of the order of 10~ 12 to 10” 8 , whereas, for a priori indepen-
dent dynasties, the typical value of A (Af, H) oscillates from 1/10 to 1/100 (and
in some rare cases, amounts to 1/1000). The sharp difference (to the extent
of several orders) between dependent and independent dynasties is obvious.
Thus, by means of A(M, if), we can reliably distinguish between dependent
and independent pairs of dynasties. The important experimental fact is that
the chroniclers have not made very “serious” mistakes; anyway, the errors are
essentially less than the value for distinguishing independent dynasties, which
permits us (within the framework of the experiment performed) to suggest a
new method for recognizing dependent dynasties and dating independent ones.
Proceeding from the analogy of the previous item; we calculate the coefficient
A (M,D) for an unknown dynasty £>, where M are known dynasties. If we
find a dynasty M for which this coefficient is small, then we can assert that
the dynasties M and D are dependent (with probability A(Af, D))> i.e., M
and D are associated with a real one whose dating is already available (for
M is dated). This method has been verified against medieval dynasties with
a priori known dating. The efficiency of the method was fully confirmed.
12.4. The frequency- damping principle. A method of ordering texts
in time
The present method permits us, for example, to discover a chronologically
correct order for individual textual chapters and duplicates on the basis of the
collection of proper names mentioned. As in the earlier methods, we strive for
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New Experimental and Statistical Methods
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creating a dating technique based on the quantitative characteristics of the
texts that are not requiring an analysis of the contents, which could be rather
ambiguous and vague.
If a document mentions any “famous” historical figures known earlier from
other, already dated chronicles, this permits us to date the described events.
However, if such an identification is not immediately successful, and if, more-
over, the events of several generations with a large quantity of historical figures
previously unknown are described, then the problem of establishing the iden-
tity of personages with those known is made more complicated. In short, we
call a fragment of a text describing the events of one generation “a generation-
chapter” . We shall assume that the mean duration of one “generation” is the
average duration of the rule of authentic kings fixed in the available chroni-
cles. This mean duration of a rule was computed by the author on the basis of
the results obtained in processing the information contained in Blair’s Tables
[6] and the GCD (see above). It equals 17.1 years. In working with authentic
texts, we face certain difficulties if we want to distinguish chapter generations
from them; therefore, we have restricted ourselves only to an approximate par-
tition of the text. Let a text X describe the events in a sufficiently large time
interval (A, B) when several generations of historical figures have replaced
each other. Suppose that X is broken into the chapter generations X(T),
where T is the ordinal number of the generation described in X(T) relative
to the numbering fixed in the text. The question arises: Have these chapter
generations been enumerated (or ordered) in the text? Or, if this enumeration
has been lost (or is doubtful), then how can it be restored? In other words,
how can the chapters be placed in time with respect to each other? As it
turns out for authentic historical texts, the following formula is valid, namely,
full name = historical figure, which means that if a time interval described
is sufficiently large (tens or hundreds of years long), then, as I verified when
analyzing a large set of historical documents, in the overwhelming majority
of cases, different personages have different full names in the same texts. A
full name can consist of several words, e.g., Charles the Bald. In other words,
the number of different persons with the same full names is negligibly small,
compared with that of all personages, which is valid for all the several hundred
texts I investigated and which describe Greece, Germany, Italy, Russia, and
others. In fact, the writer of a historical text is interested in distinguishing
between different historical figures in order to avoid ambiguity. The simplest
method to attain this is to give different full names to different people, which
is justified by calculation.
We now formulate the frequency- damping principle describing the chrono-
logically correct order of chapter generations: In the correct enumeration of
chapter generations, the author of the text, while proceeding from the de-
scription of one generation to another , also describes other historical figures ,
namely, he does not speak at all of the personages of the generations (since
they have not yet been born ) belonging to those prior to a generation numbered
T Q ; then , in describing T 0 , the author speaks of the historical figures of thts
§12 New Experimental and Statistical Methods of Dating Ancient Events 147
generation more f since the described events are related to them most; and f
finally , proceeding with the description of subsequent generations , the author
mentions the prior historical figures still less and less t since new events with
new historical figures drive out the dead .
Thus, each generation gives birth to new historical figures, who are replaced
when generations are replaced. In spite of the apparent simplicity, this princi-
ple may be useful in creating a dating method. The frequency- damping prin-
ciple can be reformulated equivalently. Since the personages are practically
uniquely determined by their full names (name = personage), we will study
the complete set of all the full names contained in a text. The term “com-
plete” will be omitted. Consider a group of names first appearing in a text
and a chapter generation T 0 . We will call them T 0 -names, and the correspond-
ing personages T 0 -personages. The quantity of all mentions (taken with their
multiplicities) of all the names in this chapter will be denoted by K(T 0 >T 0 ).
We then count how many times the same names have been mentioned in a
chapter T. The obtained number will be denoted by K(T oy T). Meanwhile, if
the same name is repeated several times, then all these mentions are counted.
Construct the graph by marking the numbers of chapters on the horizontal
scale, and K(T 0y T), where T 0 is fixed, on the vertical scale. For each T 0 , we
obtain its own graph. The frequency-damping principle is then formulated as
follows:
In the chronologically correct enumeration of chapter generations , each
graph of K(T Qy T) should vanish to the left of T 0 , attain an absolute maxi-
mum at T 0f and then gradually dampen (see Fig. 24)-
Figure 24- Ideal graph illustrating the frequency-damping principle.
We call this graph ideal. The formulated principle must be checked experi-
mentally. If it is true, and the chapters are ordered chronologically correctly,
then all experimental graphs should be close to the ideal. The experimental
verification I performed has fully fustified the frequency-damping principle.
We illustrate this with several typical examples.
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New Experimental and Statistical Methods
Chapter 3
12.5. Applications to Roman and Greek history
Example 1. Livy’s History of Rome . All the graphs of K(T 0 ,T) for the
part describing the years 750-293 B.c. and 510-293 B.c. turned out to be
almost identical with the ideal, i.e., most of the names first appearing in one
generation most often appear in the description of just this generation, and
then gradually vanish. Therefore, the principle is confirmed, and the order of
chapter generations is chronologically correct in these parts of the History of
Rome.
Example 2. Liber Pontificalis by Mommsen and Gestorum Pontificum Ro-
manorum [95]-[96]. We single out portions describing A.D. 300-560, 560-900,
900-1250, and 1250-1500. It turns out that all the graphs of K(T 0) T) practi-
cally coincide with the ideal here, too, which also confirms the principle.
We should draw our attention to one of the corollaries to the performed
experiment, namely, no fashion for names was traced in considerable time
intervals (which is not at all obvious). Certainly, some names used in antiquity
are also in use today (Peter, Mary, and so forth); however, as was made clear,
these names are either incomplete or their percentage is small in comparison
with the whole mass of dying names. The availability of the surviving names
signifies that the experimental graphs of K(T 0} T) occur in moving from left
to right, not up to zero, but up to a certain nonzero constant.
Example 3. The following set of primary sources has been taken as a text X
describing A.D. 976-1341 in Byzantine history: (1) Psellus, M., The Chrono -
graphia of Michael Psellus ; (2) Anna Comnena, The Alexiad of Anna Com-
nena ; (3) Cinnamus, Joannes, Epitome rerum ab Ioanne et Alexis ; (4) Aconi-
ates, Nicetas, Historia; (5) Acropolita, Georgius, Chronicon Const antinopoli -
tanum ; (6) Pachymeres, Georgius, De Michaele et Andronico Palaelogo, lib.
1-8; (7) Gregoras, Nichephorus, Byzantinae Historiae.
The above set contains tens of thousands of full names (taking their mul-
tiplicities into account). All the graphs K(T Qt T) turned out to be practically
identical with the ideal.
Example 4. Gregorovius’ History of the City of Rome in the Middle Ages.
The fragments describing A.D. 300-560, 560-900, and 1250-1500 were ex-
tracted from this text, each of which was divided into chapter generations,
and the total collection of the names counted several tens of thousands of
mentions. It turned out that the frequency-damping principle was also valid
and the ordering of chapters in each of these fragments chronologically correct.
A similar result was also obtained for the monograph of Kolrausch History
of Germany , in which three pieces describing A.D. 600-1000, 1000-1273, and
1273-1700 were distinguished.
Several dozen historical texts were investigated; in all the cases, the fre-
quency-damping principle was confirmed. Hence, we have a method of or-
dering textual chapter generations chronologically correctly if this order was
disturbed or unknown. Consider the totality of chapter generations in a text
§12 New Experimental and Statistical Methods of Dating Ancient Events 149
X and enumerate them somehow. For each chapter X(T 0 ) i we than plot the
graph of K(T Q ,T). All the values K(T Q ,T) with variable T Q and T can be
naturally organized into a square matrix K{T) of order n x n, where n is the
number of chapters. In the ideal theoretical case, the matrix K{T } has the
form shown in Fig. 25 and possesses zeroes below the principal diagonal and
the absolute maximum in each of its rows.
1 2 ... n
K{T }
Figure 25. The square matrix corresponding to the frequency-damping princi-
ple.
1 T 0 M n
Figure 26. Experimental graphs may not coincide with the theoretical one.
Each graph decreases monotonically and vanishes in every row. It goes with-
out saying that experimental graphs may not coincide with the theoretical one
(see Fig. 26). Now, if we alter the chapter enumeration, then K(T 01 T) will
be altered, too, because of a rather complicated redistribution of the “names
appearing for the first time”. Hence, the matrix K{T} and its elements do
vary. Changing the order of the chapters by means of various permutations
and calculating the new matrix i£{<rT}, where crT is the new enumeration
150
New Experimental and Statistical Methods
Chapter 3
corresponding to a permutation o', we will seek their order such that all or
almost all graphs will be of the form shown in Fig. 24, and the experimen-
tal matrix K{<rT} will be closest to the theoretical in Fig. 25. That order of
chapters for which the deviation of the experimental matrix from the theoret-
ical is the least should be taken as chronologically correct and required. The
description of the “proximity criterion” is omitted.
The method also permits us to date events. Suppose it is only known about
a given text Y that it depicts some events of an epoch (A, B) already described
in the chapter generations of a text X , their order being chronologically cor-
rect. How can we learn what generation precisely is described in Y if we
only make use of quantitative textual characteristics and do not consider the
possibly ambiguous contents admitting different interpretation.
The answer may be given by adjoining the text Y to the chapters of X,
treating Y as a new chapter, and ascribing a certain number T 0 to it; we then
find an optimal order of the chapters, which must be chronologically correct.
Meanwhile, we also determine a correct place for the new chapter Y. In the
simplest case, having constructed the graph of K(T 0 ,T)> we can make it as
close to the ideal as we please by placing Y appropriately. The position which
Y occupies relative to the other chapters should be taken as the required. We
thereby date the events described in Y. This technique may be also applicable
when we consider only a few names, e.g., certain “famous” ones.
Let us verify our method against texts with a priori known dating.
Example 1. Consider the period from 500 to 20 B.C. in Greek history. We
take Plutarch’s Parallel Lives as a text X. Making use of the above method,
we see that all chapter generations are placed correctly in it. As Y, we select
the Pyrrhus , describing the events usually dated to have occurred from 319
to 272 B.C. Looking for its correct position among the other chapters, we find
that it should be placed at the end of the 4th or at the beginning of the 3rd
century B.c., which is quite consistent with the earlier dating. We obtain a
rougher result, since we deal with chapters describing whole generations, and
not separate years, but then we have dated the Pyrrhus without turning to
its contents.
Example 2. We have considered and dated (by our method based on the
above-mentioned medieval Byzantine chronicles) the following Byzantine texts
describing the Crusades: X is the Histoire anonime de la premiere croisade
(Gesta Francorum et aliorum Hierosolimitanorum) whose traditional dating
to A.D. 1099 coincided with the one we obtained, namely, the end of the
11th century; and Y is Robert de Clari’s La Conquete de Constantinople ,
traditionally dated to A.D. 1204, was also related by us to the beginning of the
13th century. Thus, the efficiency of the method was confirmed in employing
medieval texts with a priori known dating.
§12 New Experimental and Statistical Methods of Dating Ancient Events 151
12.6. The frequency-duplication principle. The duplicate-discovery
method
The present method is, in a certain sense, a special case of the previous
one; however, because of its importance, we have distinguished the duplicate-
discovery method separately. Let an interval (A,B) be described in a text
X which is divided into the chapter generations X(T). Suppose they were
in general enumerated chronologically correctly, with two duplicates among
them, i.e., two chapters speaking of the same generation, and repeating each
other. Consider the simplest situation where the same chapter is repeated in
the text X twice, namely, numbered T Q and C 0 . Our method makes it pos-
sible to discover and identify these duplicates. It is clear that the graphs of
K(T q ,T) and K(C 0 >T) have the form shown in Fig. 27. The first graph ex-
plicitly does not satisfy the frequency-damping principle; hence, we have to
permute the chapters in X in order to achieve a better agreement with the
theoretical graph. All values K(C 0 ,T) vanish, since there are no “new names”
in the chapter X(C 0 ); they all have already appeared in X(T 0 ). It is obvious
that the best coincidence with the graph in Fig. 24 will be attained if we
juxtapose these two duplicates or simply identify them. Thus, if among the
chapters generally enumerated correctly, we have discovered two whose graphs
have approximately the form of those in Fig. 27, then these chapters, most
probably, are duplicates (i.e., tell of the same events) and should be identi-
fied. All the aforesaid can be transferred to the case where there are several
duplicates (three or more) .
K(C 0 ,T)
1 Tq Cq n
Figure 21. Duplicate-discovery method. The first graph does not satisfy the
frequency-damping principle.
This method has been checked against experimental material. As a simple
example, we took a Russian edition of the History of Florence by Machiavelli,
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New Experimental and Statistical Methods
Chapter 3
supplied with detailed commentary. It is evident that the commentary can be
regarded as a series of chapters duplicating the text of Machiavelli himself.
It was divided into chapter generations, which permitted us to construct a
square matrix K{T} also talcing into account the commentary. The matrix
has the form shown in Fig. 28, where the blocks filled with maxima are shown
in thick lines, which means that our method does discover duplicates which
are, in our case, made up of the commentary to the text of the History of
Florence .
12.7. Statistical analysis of the complete list of all the names
mentioned in the Bible
The following example is very important for the analysis of global chronology.
The Bible contains tens of thousands of names. It is generally known that
there are two series of biblical duplicates, namely, each generation described
in the First and Second Book of Samuel, the First and Second Book of Kings,
and then again in the First and Second Book of the Chronicles.
I have divided the Old and New Testament into chapter generations as
shown below. For reference, the canonical division into standard chapters and
verses was retained, which certainly does not coincide with the division into
chapter generations. Each of the chapter generations listed here (of course,
approximately) describes only one generation of personages, due to which the
following division of the Bible was obtained. We indicate the biblical fragments
which compose individual chapter generations. The division of Genesis into
chapter generations is given first, namely,
(1) 1-3 (Adam, Eve);
§12 New Experimental and Statistical Methods of Dating Ancient Events 153
(2) 4:1-16 (Cain, Abel);
(3) 4:17 (“Then Cain lay with his wife
(4) 4:18 (“Enoch begot Irad
(5) 4:18 (“. . . Mehujael begot Methushael . .
(6) 4:18 (“. . . Methushael begot Lantech”);
(7) 4:19-24 (“Lantech married two wives ...”);
(8) 4:25-26 (“Adam lay with his wife again”);
5:1-6 (“This is the record of the descendants of Adam”);
(9) 5:7-11 (“After the birth of Enosh ...”);
(10) 5:12-14 (“Kenan . . . lived eight hundred and forty years ...”);
(11) 5:15-17 (“Mahalalel . . . lived eight hundred years ...”);
(12) 5:18-20;
(13) 5:21-27;
(14) 5:28-31;
(15) 5:32, 6, 7, 8;
(16) 9;
(17) 10:1;
(18) 10:2;
(19) 10:3;
(20) 10:4;
(48) 10:32;
(49) 11:1-9;
(50) 11:10-12;
(51) 11:13-14;
(52) 11:15-16;
(53) 11:17-19;
(54) 11:20-21;
(55) 11:17-19;
(56) 11:24-25;
(57) 11:26-27;
(58) 11:28;
(59) 11:29-32;
(60) 12;
(61) 13;
(62) 14-24;
(63) 25:1-2;
(64) 25:3;
(65) 25:4;
(66) 25:5-10;
(67) 25:11-18;
(68) 25:19-26;
(69) 25:27-34;
(70) 26-33;
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New Experimental and Statistical Methods
Chapter 3
(71) 34-36;
(72) 37, 38;
(73) 39-50.
Genesis ends here.
(74) Exodus; (75) Leviticus; (76) Numbers; (77) Deuteronomy; (78) The
Book of Joshua; (79) The Book of Judges 1; (80) The Book of Judges 2; . . . ;
(96) The Book of Judges 18; (97) The Book of Judges 19-21; (98) Ruth; (99)
The First Book of Samuel 1—15; (100) The First Book of Samuel 16—31; (101)
The Second Book of Samuel; (102) The First Book of Kings 1-11; (103) The
Fir^Book of Kings 12; (104) The First Book of Kings 13;...; (112) The
First Rook of Kings 22; (113) The Second Book of Kings 1; (114) The Sec-
ond Book of Kings 2,...; (135) The Second Book of Kings 23; (136) The
Second Boole of Kings 24, 25; (137) The First Book of the Chronicles 1—10;
(138) The First Book of the Chronicles 11-29; (139) The Second Book of
the Chronicles 1-9; (140) The Second Book of the Chronicles 10;. .. ; (166)
The Second Book of the Chronicles 36; (167) The Book of Ezra; (168) The
Book of Nehemiah; (169) Esther; (170) The Book of Job; (171) Psalms; (172)
Proverbs; (173) Ecclesiastes; (174) The Song of Songs; (175) The Book of the
Prophet Isaiah; (176) The Book of the Prophet Jeremiah; (177) Lamentations;
(178) The Book of the Prophet Ezekiel; (179) Daniel; (180) Hosea; (181) Joel;
(182) Amos; (183) Obadiah; (184) Jonah; (185) Micah; (186) Nahum; (187)
Habakkuk; (188) Zephaniah; (189) Haggai; (190) Zechariah; (191) Malachi;
(192) Matthew; (193) Mark; (194) Luke; (195) John; (196) Acts of the Apos-
tles; (197) A Letter of James; (198) The First Letter of Peter; (199) The
Second Letter of Peter; (200) The First Letter of John; (201) The Second
Letter of John; (202) The Third Letter of John; (203) A Letter of Jude; (204)
The Letter of Paul to the Romans; (205) The First Letter of Paul to the
Corinthians; (206) The Second Letter of Paul to the Corinthians; (207) The
Letter of Paul to the Galatians; (208) The Letter of Paul to the Ephesians;
(209) The Letter of Paul to the Philippians; (210) The Letter of Paul to the
Colossians; (211) The First Letter of Paul to the Thessalonians; (212) The
Second Letter of Paul to the Thessalonians; (213) The First Letter of Paul
to Timothy; (214) The Second Letter of Paul to Timothy; (215) The Letter
of Paul to Titus; (216) The Letter of Paul to Philemon; (217) A Letter to
Hebrews; and (218) The Revelation of John.
Thus, the Old Testament has been broken into 191 chapter generations.
The New Testament consists of Chapters 192-218. We shall now consider only
Chapters 1-170. Meanwhile, Genesis 4:18 was divided into three generation
chapters. The other subsequent chapters (including the New Testament) will
not be required for the present and are therefore omitted.
In 1974-1979, V.P. Fomenko and T.G. Fomenko carried out the enormous
job of compiling the complete list of all the names mentioned in the Bible,
taken with their multiplicities, and exactly distributing the names in genera-
tion chapters. It turned out that altogether about 2,000 names are mentioned,
whereas, the number of references with their multiplicities amounts to tens of
§12 New Experimental and Statistical Methods of Dating Ancient Events 155
thousands. Accordingly, all the graphs of K(T 0i T) were constructed, where
the number T ranges over all the chapters listed. The graphs constructed for
the chapters from the First and Second Book of Samuel and the First and
Second Book of Kings turned out to have the form of the graph in Fig. 27,
i.e., names first appearing in these chapters are then given “birth” again in
the corresponding chapters from the First and Second Book of the Chronicles.
The corresponding part of the matrix K{T} is shown in Fig. 29. Two parallel
diagonals filled with the absolute maxima of the rows are marked with two
thick lines. Thus, our method has successfully discovered and identified those
duplicates in the Bible which were also known earlier as such.
K{T)
Figure 29. Part of the square frequency matrix for the Bible. Duplicates: (1-2
Samuel -f 1-2 Kings) and 1-2 Chronicles.
12.8. Statistical analysis of the complete list of all parallel passages
in the Bible
The application of these methods is sometimes made easier by the fact that,
for many a historical text, the commentators have revealed the repeating frag-
ments. By a “repetition”, we mean not only a repetition of a name, but also a
repeated description of a particular event, and so forth. For example, the same
descriptions, lists of names, religious formulas, and so on, are encountered in
the Bible many times. All of them have been discovered in the Bible long ago
and were systematized and mentioned in the list of parallel passages; namely,
close to certain verses, it is indicated which verses of this or other books of
the Bible are regarded as “repetitions” (which are “parallel”). If a text X
under investigation has been supplied with this, or a similar, list of parallel
passages, then we can apply the duplicate-discovery method, regarding the
repeated fragments as “repeated names” .
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New Experimental and Statistical Methods
Chapter 3
We illustrate this with an example. Consider the books of the Bible from
Genesis to the Book of Nehemiah. The partition of the Bible into chapter
generations has been given above. We now enumerate them in the order in
which they replace each other in the canonical ordering. The list of “parallels”
contains approximately 20,000 repeated verses. We count in each X(T) the
number of verses which have not yet appeared in the preceding chapters X(T)
(i.e., first appearing in X(T 0 )). We denote their number by II(T 0 ,T 0 ). We
then count how many times these verses repeat in the subsequent X(T). We
denote the obtained quantities by n(T 0 ,T). All graphs of n(T 0 ,T) have been
constructed (altogether 169 graphs). They differ from those of K(T Q , T) only in
dealing with verses instead of names, and names instead of verses. The verses
which are not repetitions of each other or some other verse are regarded as
“different names”. 4
Hence, in the correct chronological order of chapter generations, with du-
plicates being absent, the verse repetition graphs n(T 0 ,T) should have the
form of that shown in Fig. 24. Similarly to the case of employing names, the
author of a text (for the correct order of the described events) says nothing of
the events of the generation T 0 in the preceding chapters (these events have
not yet occurred), says a lot while describing the events of the generation
T 0 , and recalls them still less and less in the subsequent chapters, i.e., the
graph possesses an absolute maximum at the point T 0 , vanishes to the left
of T 0 , and decreases monotonically to the right of T 0 . An experimental check
which I performed has justified the frequency-damping principle for all the
fragments of the Bible, listed below, namely, (1) Genesis 1-5; (2) Genesis 6-
10; (3) Genesis 11; (4) Genesis 12-38; (5) Genesis 39-50; Exodus; Leviticus;
Numbers; Deuteronomy; The Book of Joshua; The Book of Judges 1-18; (6)
The Book of Judges 19-21; Ruth; The First and Second Book of Samuel; The
First and Second Book of Kings 1-23; (7) The First and Second Book of the
Chronicles; The Book of Ezra and Nehemiah. It turns out that all the graphs
of n(T 0 , T) are of the form of the theoretical graph in Fig. 24 for each of these
texts, which means that the frequency-damping principle is valid in this case;
besides, the order of the chapters is chronologically correct, while duplicates
are absent.
If all the chapters of a text are enumerated correctly as a whole, then
the duplicates can be discovered by plotting the “verse repetition graphs” of
n(T OJ T). If two chapters X(T 0 ) and X(C Q ) are duplicates, then the graphs of
n(T 0 ,T) and n(C , OJ T) are of the form demonstrated in Fig. 27. This method
has also been experimentally checked for the above example, namely, [the
First Book of Samuel -f the Second Book of Samuel + the First Book of
Kings + the Second Book of Kings] duplicate [the First and Second Book
of the Chronicles]. The construction of the graphs of n(T 0 ,T) revealed that
the duplicates were just those chapters from [the First and Second Book of
4 This entire enormous job was carried out by V.P. Fomenko and T.G. Fomenko.
§12 New Experimental and Statistical Methods of Dating Ancient Events 157
Samuel 4* the First and Second Book of Kings] and [the First and Second
Book of the Chronicles], which were duplicates from the standpoint of the
graphs of K(T Qy T ), thus indicating the complete consistency of the results of
the application of both methods. Meanwhile, it should be noted that the list
of parallelisms is not identical with that of repeated names, since, for example,
many fragments (verses of the Bible) not containing names at all axe regarded
to be “parallel” .
12.9. Duplicates in the Bible
We now continue with a short description of the results of applying our meth-
ods to the antique and medieval chronological data. In doing so, duplicates
traditionally regarded as different, and dated today to substantially different
years, have been discovered unexpectedly.
We have applied the duplicate-recognition method (on the basis of the
graphs of K(T 0i T) and II(T 0 ,T)) to the Bible, namely the Old Testament
from Genesis to Esther. The obtained results will be represented as a line (=
chronicle) B in which duplicates (i.e., fragments of the Bible speaking of the
same events, as follows from the verification described above of the frequency-
duplication principle) have been denoted by identical symbols (letters). Thus,
the line
B — TKTHTKTKTHTTPT C a ,
n
P
which means that the entire “historical part” of the Old Testament consists
of several fragments, namely, T, K,H, II, P, with some of them repeating
several times, and placed differently in the Canon, thus yielding the above
line B. In other words, many of the fragments (indicated in B) of the Old
Testament actually describe the same events, thus contradicting the tradi-
tional ideas according to which the different books of the Bible (except the
First and Second Book of Samuel + the First and Second Book of Kings and
the First and Second Book of the Chronicles) describe different events. Let
us decode the symbols of the chronicle B. Indicating a symbol, we list the
corresponding fragments of the Bible in brackets. Thus, B equals T (Genesis
1-3) K (Genesis 4-5) T (Genesis 6-8) H (Genesis 9-10) T (Genesis 11:1-9)
K (Genesis 11:10-32) T (Genesis 12) K (Genesis 13-38) T (Genesis 39-50,
Exodus) H/U/P (Leviticus, Numbers, Deuteronomy, The Book of Joshua and
the Book of Judges 1-18) T (The Book of Judges 19-21) T (Ruth, the First
and Second Books of Samuel, the First Book of Kings 1-11) P (the First
Book of Kings 12-22, the Second Book of Kings 1-23) T (the Second Book of
Kings 24) C a (the Second Book of Kings 25, the Book of Ezra, the Book of
Nehemiah, Esther).
158
New Experimental and Statistical Methods
Chapter 3
1 10 20 30 40 SO 00 70 80 90 100 110 120 130 140 190 160 170 160 190 200 210 216
T T T T T TT TT T
Figure 30a . Square matrix of biblical names (detailed diagram). The most
essential concentrations (duplicates) are shown.
Besides, the sequence TPTC a (at the end of the line B) was also described
in the First and Second Book of the Chronicles. The two latter series of
duplicates, which were known before, are unique. The other duplicates exhib-
ited above were unknown earlier and were discovered on the matrix K{T}
§12 New Experimental and Statistical Methods of Dating Ancient Events 159
Figure 30b. Part of the square matrix of biblical names (rough diagram).
of the biblical Chapters 1-170 as follows. Two earlier-known duplicate series,
(namely, Chapters 98-137 and their duplicates, Chapters 138-167) imply that,
along with the maxima filling the principal diagonal, there is another diagonal
also composed of maxima and parallel to the principal in rows 98-137 (see
Figs. 30(a), 30(b)). These diagonals are represented as black oblique segments.
Rows 138-167 practically consist solely of zeroes.
The other duplicates could be discovered by approximately identical local
peaks placed in the intersections of the corresponding rows and columns.
The duplicates of the series T, as encountered in the Old Testament most
frequently, are represented in Fig. 30a, b (see the particulars below).
12.10. The enquete-code or formalized “biography” method
Ancient literature used to resort to cliches and borrowings, for example, in
the description of some particular rulers. Sometimes, the chroniclers ascribed
the character and deeds of ancient kings to other rulers. To detect and study
these cliches and duplicates, I have introduced the concept of enquete-code,
or formalized “biography” . An authentic ruler described in the chronicles will
thereby acquire a “historical literary biography” , which might have nothing to
do with the authentic one and can be legendary. On the basis of the investiga-
tion of a large number of historical biographies, a table called an enquete-code
was worked out, hierarchically ordering the facts of the biography with the
decrease of their invariance relative to the personal attitudes of the authors.
160 New Experimental and Statistical Methods Chapter 3
An enquete-code consists of 34 items, each of which contains several subitems,
see Chapter 4.
How can we know, without resorting to the analysis of the contents of
the chronicles, whether or not these chronicles describe the same authentic
dynasty? If the rule durations are indicated in the chronicles, then we can
apply the numerical dynasty recognition method (see Section 12.3 above).
However, if such numerical data are not available, then the problem gets more
complicated. Thus, how can the same authentic dynasty be distinguished from
the set of all enquete-codes? To solve the problem, a method based on the
small- distortion principle (see above) has been worked out. In this case, it is
formulated thus:
If the enquete-codes of two dynasties differ “little” from each other, then
they represent the same authentic dynasty; however, if two enquete-codes rep-
resent different dynasties, then they are “far” from one another.
Here, we omit the description of the numerical coefficient similar to A (Af, H)
and permitting us to separate reliably the “dependent enquete-codes” from
the “independent” ones (see the particulars in [2]). The experimental check
has confirmed the validity of the small-distortion principle in this case, too: It
turned out that the enquete-codes representing the same dynasty were much
less different from each other than those of different authentic dynasties. It
is obvious that this circumstance permits us to date the enquete-codes of
dynasties in accordance with the above procedure.
12.11. A method for the chronological ordering of ancient maps
I also developed a method for the chronological order of ancient maps. Any ge-
ographical map reflects the state of geoscience of the period when the map was
made. With the development of scientific ideas, maps improve more and more,
i.e., the amount of incorrect information decreases, and that of reliable data
increases. An optimal map-code permitting us to represent any map (given
graphically or described verbally) as a table similar to an EC was worked
out and based on the analysis of concrete ancient maps. The list of items of
this table will be omitted here. An experimental check performed in 1979—
1980 permitted us to formulate and justify the following “chart-improvement
principle” .
If a chronologically correctly ordered sequence of maps is given, then, in the
transfer from old charts to new ones, two processes occur, namely, incorrect
features which do not correspond to real geography vanish and no longer appear
on the maps, i.e., “errors are not repeated”, and the correct ones which have
been introduced, e.g., availability of a channel or river, profile of the bank, are
fixed and retained in all subsequent charts.
Due to the role which maps always played in sea-faring, and so forth, this
chart-improvement principle has been introduced because of the urgent re-
quirements arising from practice. The principle has been verified according to
the procedure used in the previous items. We establish a certain order of the
§12 New Experimental and Statistical Methods of Dating Ancient Events 161
charts, construct the graph of L(T 0 ,T) for each T 0 where L(T 0 ,T 0 ) equals the
number of features first appearing on a map T 0i whereas L(T Qi T) indicates
how many of them remained on the chart T. We should assume an ordering of
the charts to be chronologically correct if all the graphs of L(T 0 ,T) are close
to that represented in Fig. 24 (and incorrect otherwise). In particular, charts
which seem to be visually close turn out to be close in time, too: Each epoch
is characterized, as can be seen, by its unique set of maps. The verification of
the principle has been made more complicated by the fact that few ancient
charts survived up to the present time. Nevertheless, a sufficient number of
charts permitting us to verify the model were collected.
Those of the 3rd-4th century turned out to be quite primitive and rather
far-fetched; their quality then improves steadily until, in the 16th century,
we encounter sufficiently correct maps, and even globes dating from the 17th
century. Meanwhile, the quality improved extremely slowly. For example, the
geographical knowledge of 16th-century Europe was still very far from that
of today. One map signed in 1522 by T. Occupario represented Europe and
Asia in proportion sharply contrasting to the modern Europe: Greenland
is a European peninsula there, Scandinavia elongated into a thin strip, the
Bosphorus and Dardanelles greatly extended, the Black Sea distorted ver-
tically, the Caspian Sea drawn in horizontal direction and literally made
unrecognizable, and so forth. The only region reflected more or less correctly
is the Mediterranean; still, Greece is represented as a triangle without the
Peloponnese. The ethnographical data on this and other maps of the time
are yet much farther from those fixed by traditional history. For example,
Dacia and Gottia (land of the Goths?) are placed in Scandinavia, Albania on
the Caspian Sea, and China is completely absent, the Judei are in Northern
Siberia, and so forth. A map of Cornelius Niccolai (1598) abounds in similar
distortions, but now to a lesser degree. Finally, a globe of the 17th century in
the Moscow History Museum already reflects the true geographical position
quite well.
The method permits us to date maps, including “ancient” ones, according to
the procedure described above. The obtained results may be quite unexpected.
We illustrate this with some typical examples.
(1) The famous chart from the Geography by Ptolemy, ed. Basileae, 1545,
which is regarded today as ancient, does not fall into the 2nd century, but the
15-16th century, i.e., the time when Ptolemy’s book was published. This fact
makes us recall quite a similar situation with the Almagest (see above).
(2) The no less famous ancient chart Tabula pentingeriana (see e.g., [1],
pp. 232-233, Fig. 48) falls not at the turn of the millennium, i.e., time of
Augustus, but into the 1 1— 12th century. The divergence from the traditional
dating is more than 1,000 years.
(3) Series of ancient maps (which are, though, later reconstructions from
verbal descriptions in ancient texts; see [75]) by Hesiod (dated traditionally
to the 8th century B.C.). Hecataeus (6-5th century B.C.), Herodotus (5th
century B.C.), Democritus (5-4th century B.C.), Eratosthenes (276-194 B.C.),
162 New Experimental and Statistical Methods Chapter 3
the “globe” of Crates (168-165 B.c.) if dated by the above method, then fall
into the 7- 13th century.
§13. Construction of the Global Chronological Diagram and Certain
Results of Applying the Dating Methods to Ancient History
13.1. The “textbook” of ancient and medieval history
In 1974-1980, 1 carried out the analysis of the global chronology of the ancient
and medieval history of Europe, the Mediterranean, Egypt, and the Near East.
The historical and chronological data of J. Blair’s [6] and 14 other tables
(see above) were completed with the information from 222 texts, chronicles,
annals, and others, containing together the description of practically all basic
events occurring in the indicated regions from 4000 B.c. to A.D. 1800 if dated
traditionally. All this information (wars, kings, basic events, empires, etc.)
was then represented graphically as the global chronological diagram (GCD)
constructed on the horizontal “time” axis. Each epoch with all its events was
represented in detail by the lists and dates at the corresponding place on the
time axis.
We then applied the dating of events and duplicate-recognition methods de-
scribed above and in [2]— [5] and, in particular, computed the values d(X,Y)
for different pairs of surveying historical texts X ) Y embracing large time inter-
vals. The quantity A (M, H ) for the different dynasties M, H from the GCD,
the coefficients L(P, i/), and values measuring the enquete-code proximity
were also calculated. This extensive experiment unexpectedly led to the dis-
covery of pairs of epochs regarded as independent by traditional history, but
for which the coefficients d(X, Y), A etc., turned out to be extremely
small and characterizing necessarily dependent epochs, texts, dynasties or
enquete-codes. We illustrate this with an example.
13.2. Duplicates
It was discovered that the history of ancient Rome in 753-236 B.c. overlapped
with that of the medieval one in A.D. 300-816. More precisely, the epoch (A, B)
from 300 to 816 was described, for example, in the fundamental work of F.
Gregorovius, History of the City of Rome in the Middle Ages ; the epoch (C, D)
from the year 1 to 517 since the foundation of Rome (which occurred, as is
thought today, in 753 B.c.) is described in the following two texts.
The History of Rome by Livy from the year 1 up to 459 since the foundation
of Rome; Livy’s text breaks off at that point, and the other books are lost.
Therefore, the end of the period (C, D) from the year 459 to 517 since the
foundation of Rome was “covered” by the monograph of V.S. Sergeev Essays
on the History of Ancient Rome by extending Livy’s text. Meanwhile, we
have based ourselves on the discovered strong correlation of Sergeev’s text
with that of Livy with the proximity coefficient d = 2 • 10“ 12 (see above and
Fig. 20).
§13 Construction of the Global Chronological Diagram 163
The computation of d(X,Y ), where X is the text of Gregorovius (medieval
Rome), and Y the sum of Livy’s and Sergeev’s (ancient Rome) texts, yields
d(X, Y) = 6 • 10” 11 . However, if we drop Sergeev’s text and compare the text
X 1 = part of Gregorovius’ text from 300 to 75& and the text Y* = part of
the History of Rome by Livy from the year 1 to 459 since the foundation of
Rome, then we can compute that p(X , 1 Y t ) = 6 • 10~ 10 . Both results indicate
the dependence of the two epochs described in the modern textbook, namely,
the antique and the medieval one; more precisely, dependence of the primary
sources describing them (on which all the later texts are based). This depen-
dence is expressed vividly and is of the same character as that between the
texts describing the same events (see Fig. 31).
+300 +758 +816
Figure 31. Correlation between volume functions for Livy y s “History of Rome”
and the “History of the City of Rome in the Middle Ages ” by Gregorovius.
All such epochs (A, B ) and (C, D) that are anomalously close from the
standpoint of the coefficient d(X,Y) have been marked on the GCD. We call
such epochs cf-dependent and represent them by the same symbols.
13.3. Dependent dynasties
An independent experimental investigation of the GCD was then also carried
out on the basis of the dependent dynasty recognition method, for which lists
of all the rulers from 4000 B.c. to A.D. 1800 for the indicated regions have
been made (see [6]). The method described in Section 3 was applied to this
collection (each of the dynasties consisting of 15 kings). The experiment has
unexpectedly led to the discovery of the special pairs of dynasties M and H
earlier regarded as independent (in all respects), but for which the proximity
coefficient A(Af, H) is of the same order as for a priori dependent dynasties,
i.e., it oscillates from 10~ 12 to 10“" 8 .
Let us give several examples. By a dynasty, we understand a sequence of
actual rulers of a country without regard to their titles and relations. Due to
164
New Experimental and Statistical Methods
Chapter 3
the existence of co-rulers, difficulties in arranging dynasties in a row some-
times arise; the simplest principle of their ordering, i.e., with respect to the
midpoints of the periods of their rules, has therefore been adopted. We call
a sequence of rules in the history of a state a dynastic stream , and its subse-
quences obtained by rejecting some particular co-rulers dynastic jets. It was
required of a jet that it should be monotone (i.e., the midpoints of the periods
of the rules making up a jet should be monotonically increasing) and complete
(i.e., gap-free and covering the whole period emhraced by the stream, with
overlapping being permitted). Understandably, these requirements may not
always be satisfied in real situations. For example, a chronicler’s story may
omit the year between two rules, and so forth. Therefore, we should admit
insignificant gaps (no more than one year long) and the three types of errors
described and modelled in [5]).
There is another reason for which a precise formal picture can be distorted:
It is sometimes difficult to establish with certainty the start of a rule (e.g.,
to count from the moment of actually starting the rule or formally ascending
the throne), whereas there are usually no difficulties about its end: In most
cases, it is the ruler’s death. For example, sources supply different dates for the
enthronement of Frederick II, namely, 1196, 1212, 1215 and 1220, thus making
it necessary to w double” the king or to consider him even in a larger number of
versions, all included in the general dynastic stream. Meanwhile, it was required
that no jet should contain two different variants for the same ruler.
We now give examples of dependent dynasties.
(1) M = Roman Empire actually founded by Lucius Sulla in 82-83 B.c. and
ending with Caracalla in A.D. 217; H = Roman Empire restored by Lucius
Aurelian in A.D. 270 and ending with Theodoric in A.D. 526 (A (M y H) =
10~ 12 ); the dynasty M is obtained from H by shifting the latter backwards
by ca. 333 years).
(2) M = Dynasty of kings of Israel in 922-724 B.C., described in the First
and Second Book of Samuel and the First and Second Book of Kings; H =
Jet from the Roman Empire in A.D. 300-476 (A (M,H) = 1.3 • 10~ 12 ).
(3) M = Dynasty of the kings of Judah in 928-587 B.C., described in the
First and Second Book of Samuel, and the First and Second Book of Kings;
H = Jet of the Eastern Roman Empire in A.D. 300-552 (A(Af, H) = 1.4 •
io- 12 ).
All the pairs discovered by our method turned out to be close to those given
in [1], though ours are sometimes substantially different (especially in the case
of the third dynasty) from those suggested there on the basis of a simple
selection. The fact that the three pairs of [1] did not turn out to be optimal
in the sense of A (M, H ) is related to Morozov’s basing his conclusions on
the “visual similarity” of the dynasty graphs. Our analysis has demonstrated
that several dozen such and even more “outwardly similar” and necessarily
independent pairs of dynasties can be exhibited; hence, the problem arose
of finding a quantitative method for separating dependent pairs from clearly
independent ones.
§13
Construction of the Global Chronological Diagram
165
All the other pairs of dependent dynasties listed below, and also those
indicated in the GCD, were unknown earlier, and I discovered them while
investigating the material of the GCD by means of the above methods.
(4) M = Dynasty of popes in 140-314; H = Dynasty of popes in 324-532;
A (M,H) = 8.66 • 10“ 8 . This pair is perfectly consistent with pair 1.
(5) M = Empire of Charlemagne from Pippin of Heristal to Charles the
Fat, i.e., in 681-887; H = Jet of the Eastern Roman Empire in 324-527,
= 8.25 • 10“ 9 .
(6) M = Holy Roman Empire in 983-1266; H = Jet of the Roman Empire
in 270-553; A (A/, if) = 2.3 • 10~ 10 . The dynasty H is obtained from M by
shifting the latter backwards by ca . 720 years.
(7) M = Holy Roman Empire in 911-1254; H = German Roman Empire
of the Habsburgs in 1273-1637(1); A(A/,if) = 1.2 • 10“ 12 . The dynasty M
is obtained from H by shifting the latter backwards by 362 years as a solid
block.
(8) M = Holy Roman Empire in 936-1273; H — Roman Empire from 82-
217; A 1.3 KT 12 .
(9) M = Dynasty of the kings of Judah in 928-587 B.c. (First and Second
Books of Samuel and First and Second Books of Kings; see also Pair 3); H =
Jet of the Holy Roman Empire in 911-1307(1), A (M y H) = 10” 12 .
(11) M = Dynasty of the kings of Israel in 922-724 B.c. (First and Second
Books of Samuel and First and Second Books of Kings); H = Formal dy-
nasty of the Roman coronations of German emperors in Italy in 920—1 170(1);
A {M y H) = 10“ 8 , meaning the Roman coronations of the emperors of the
Saxon, Salian, Frankish, and Swabian (of Hohenstaufens) of the German dy-
nasties.
The two latter pairs are especially startling, since they signify an overlap-
ping of the history in the Old Testament with the medieval Roman-German
history in the 10th-14th century. This overlapping differs by ca. one thousand
years from that suggested in [1], and by two thousand years from traditional
chronology.
Other examples of special dynasty pairs are demonstrated in the GCD (see
below). Thus, for example, we cannot help stressing the striking overlapping
of the history of medieval Greece in 1250-1460 with part of the history of
ancient Greece in 510-300 B.c.
13.4. The agreement of different methods
After all dynasty pairs in the GCD have been investigated, all the special
(dependent) dynasty pairs M and H , such that A(Af , H) is of the same order as
for certainly dependent dynasties, i.e., from 10“ 8 to 10~ 12 , have been denoted
by identical symbols. The GCD with this additional structure turned out to
coincide with the one on which all pairs of epochs proximal in the sense of the
coefficient cf(X, Y) have been marked. An extremely important fact is valid,
namely, the application to the GCD of all dating methods worked out leads to
the same result: Though obtained by essentially different methods, the dates
166
New Experimental and Statistical Methods
Chapter 3
are consistent. In particular, epochs close in the sense of d(X , Y ) are also close
in the sense of A (Af, H ), and also in the sense of the coefficient measuring the
proximity of the dynastic enquete-codes. Moreover, the obtained results are
consistent with astronomical dating; they are also fully consistent with the
forward shift of "ancient” eclipses, discovered in [1],
By way of example, we describe part E of the GCD from 1600 B.c. to
a.d. 1700 for Italy, Germany, and Greece. The result is given as the line
£, in which historical epochs have been denoted by letters. Here duplicates
("repetitions”), i.e., epochs duplicating each other (i.e., close in the sense of
our methods) are denoted by identical letters. Due to the very extensive data,
we give only a rough sketch; (see also Fig. 32 with a time scale).
This chronicle line E (part of the "modern textbook” GCD) contains obvi-
ously repeating duplicate epochs and can be resolved into a simple composi-
tion of three shifts of four practically identical chronicles. Thus, we can write
schematically that E = C\ + C 2 + C3 + C4.
13.5. Three basic chronological shifts
It is important that all four lines, each of which represents a certain chronicle
made up of fragments of the "modern textbook” GCD, are almost identical.
One of the explanations of this basic result we obtained can be the fact
that the "modern textbook” of ancient and medieval European history is a
“fibred” chronicle derived by gluing together the four practically identical
replicas of the shorter chronicle C\. The other three chronicles C2, C3, and
C 4 are obtained from C\ by shifting it as a rigid block backwards by 333,
1053, and 1778 years. In other words, the entire "modern textbook” can be
completely restored from its lesser parts C\ or C 0 wholly placed to the right
of A.D. 300. Moreover, practically all information in the chronicle lines C 0
and C\ turns out to be situated to the right of A.D. 960, i.e., each epoch of
history, placed to the left (below) of A.D. 960, is a "reflection” of a certain
later historical epoch wholly placed to the right of A.D. 960, and which is the
"original” of all the duplicates generated by it. The fragments K, H, and II
of the original line C Q contain very little information, and the principal part
of C 0 is concentrated in the fragments P, M , T, and C placed to the right of
a.d. 920-960.
The principal result of the research I carried out in 1974-1980 is that this
assertion is valid not only for the line E (reflecting the history of Europe),
but also for the whole GCD.
Recall that the epochs designated by the same symbols are duplicates, i.e.,
consist of the “same” events. For example, this can also be applied to the
following wars, namely, the Trojan war, the war with the Tarquins in Rome,
the war between Sulla, Pompey, and Julius Caesar in Italy, the Gothic war
in the middle of the 6th century A.D. in Italy, the war in Italy in the middle
of the 13th century (fall of the Hohenstaufen dynasty, establishment of the
House of Anjou). This latter war and the fall of Constantinople in 1204 (the
§13
Construction of the Global Chronological Diagram
167
Figure 32. Global chronological diagram of the ancient and medieval world.
Three chronological shifts.
168 New Experimental and Statistical Methods Chapter 3
Fourth Crusade) are probably the “original” of all the other wars denoted on
E by the symbol T.
In my opinion, the resolution discovered in the GCD into the sum of the
three shifts can be explained naturally. The process of creating the global
chronology and history of ancient times started in the late Middle Ages: The
historical data accumulated until that time was ordered for the first time,
namely, it was separated into chronicles, annals, and so forth; however, while
“patching” all these pieces together into a unique diagram, an error was made,
namely, the four replicas of the same chronicle ( C\ or C Q ; see above), in general
describing the same history of Europe and the Mediterranean, were regarded
by the chronologists as being different, i.e., speaking of different events, due
to which they were “patched” together not in a parallel fashion as it should
have been done, but in series, with shifts of 333, 1053, and 1778 years.
The “shorter chronicle” C\ has thus been converted into the “longer chron-
icle” Ey i.e., the “modern textbook” of ancient and medieval history. I made
clear the reasons which might have led to such a confusion and which gen-
erated just these and not any other shifts. Since the analysis of the material
requires lengthy digressions into history and is far outside the framework of
the present treatise, it is omitted here (for details, see Vol. 2).
13.6. Biblical history and European history
13.6.1. Volume graphs for the Old Testament and the “European
textbook” for 850 B.C. to A.D. 1400. The “modern textbook” GCD
also possesses other portions that differ from E , contain duplicates and are
resolvable into the sum of several “shifted chronicles” . This can be applied to
biblical history, too. We have already mentioned above that the Bible contains
many duplicates (see the line B).
It was not accidental that in the description of B we had used the same
letter symbols as for the “European” line E. As a matter of fact, B turns
out to be completely coincident (identical) with the part of Ey describing
Roman-Greek-European ancient and medieval history. More precisely,
E = T I< T H T (line B)
PCP n C
C
P
This overlapping of B and part of Ey if the time scale is taken into account,
is shown in Fig. 32.
It can be seen that the line B (Old and New Testament) overlaps with the
part of the “European textbook” E from 850 B.C. to A.D. 1400./ However,
since the Bible contains many duplicates, the entire Old Testament as well
§13
Construction of the Global Chronological Diagram
169
as the “modern textbook” E can be completely restored from its lesser part,
namely, be placed to the right of a.d. 300.
Moreover, in reality, practically all of the Old Testament, just as the whole
Bible and line E , can be restored from its part describing the events tradi-
tionally dated by 960-1400.
Meanwhile, the New Testament probably describes the events occurring
in the 11th century in Italy. It follows from the structure of the discovered
duplicates that, in particular the epoch of Jesus, the turn of the millennium
is a duplicate of the “epoch of Hildebrand”, the famous Gregory VII (11th
century). It is precisely “Hildebrand’s epoch” that opens the Crusaders’ era
marked by the famous separation of the churches in A.D. 1054 and begins the
new, Reformation Church in Europe.
Overlapping of the line B (Old Testament) with part of line E was obtained
by me by formally applying the above methods. We shall demonstrate this by
the volume graphs compared by means of the coefficient d(X, Y) (see above).
Consider the interval from 800 B.C. to A.D. 1300 in Italian and European
history. We take the sum of two fundamental monographs, namely, Niese
[14] from 800 B.c. to a.d. 552 and Gregorovius [53] from a.d. 300 to 1300.
Combining the two sources, we obtain a text X now describing the whole
interval (A,B). We then break X into the union of fragments X(T ), which
permits us to construct the volume graph for the chapters X(T) in the whole
interval from 800 B.c. to a.d. 1300, i.e. 2,100 years long.
We now consider the Old Testament (Chapter generations 1-170). The
Chapter volume graph should be constructed and compared with the cor-
responding graph for X , the difficulty being that the Bible has no time scale
which would be sufficiently detailed. However, as has already been indicated,
the Bible admits a practically unique decomposition into the chapter gen-
erations B(T ), where the ordinal number T varies from 1 to 218. Consider
the first 137 chapter generations, i.e., from Genesis to the Second Book of
Kings. Since the First and Second Books of Samuel and Kings duplicate the
First and Second Books of Chronicles, Chapters 138-167 duplicate Chapters
98-137; therefore, we are not interested in them now. Chapters 103-137 have
been described in the First and Second Books of Kings with detailed chrono-
logical indications, which permits us to determine the length of the described
time interval to rather considerable accuracy: It is 341 years (see also how the
same interval has been measured in [7]). Such circumstantial chronological
information for the remaining Chapters 1-102 is absent in the Bible; hence,
to find the time interval described, we have to do it in a rather rough manner.
The analysis of Chapters 1-102 has shown that while describing the events
of one generation, almost every one relates the latter to some principal his-
torical figure, i.e., a “ruler”, whose “rule” can be taken as the “duration” of
the generation. We have already said earlier that the mean duration of a rule
equals 17.1 years in the whole of ancient and medieval history fixed in written
sources, i.e., approximately 17 years. Basing ourselves on this mean value, we
obtain that the period “covered” by 102 generations can be estimated to be
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New Experimental and Statistical Methods
Chapter 3
approximately 102-17 = 1734 years long. Thus, we can assume that the period
described in the “historical part” of the Old Testament (without moralistic
texts) is 1734 + 341 = 2075 years. We see that this figure turns out to be
extremely close to 2,100 years, i.e., the period described in the text X (see
above). Therefore, we can compare the chapter volume graphs for X and the
Old Testament, for which we have to refer the texts to one and the same scale.
The simplest of them may be the partition of the whole (A, B) = 800 B.c.
to A.D. 1300 into 19 line segments, arising if we single out in (A, B) all the
epochs T discovered above while analyzing the GCD.
The bounds of the obtained 19 segments are 800, 770, 750, 520, 509, 380, 100
B.c., and a.d. 14, 98, 235, 305, 493, 552, 715, 901, 1002, 1054, 1250, 1268 and
1300. Since the line segment (A, B) may overlap, because of the same length,
with the period described in the Old Testament, we obtain the corresponding
division of the Chapter-generation sequence 1-170 in B into the following 19
groups, namely, the period from 800 to 770 B.C. is not described, that from
770 to 750 B.c. corresponds to Chapter Generation 1, from 750 to 520 B.c.
to Chapters 2-14, from 520 to 509 B.c. to Chapter 15, from 509 to 380 B.c.
to Chapters 16-23, from 380 to 100 B.c. to Chapters 24-39, from 100 B.c.
to a.d. 14 to Chapters 40-46, from a.d. 14 to 98 to Chapters 47-50, from
a.d. 98 to 235 to Chapters 51-59, from a.d. 235 to 305 to Chapters 60-62,
from a.d. 305 to 493 to Chapters 63-73, from a.d. 493 to 552 to Chapters 74-
78, from a.d. 552 to 715 to Chapters 79-88, from a.d. 715 to 901 to Chapters
89-97, from a.d. 901 to 1002 to Chapters 98-102, 141, 142, from a.d. 1002 to
1054 to Chapters 143-147, from a.d. 1054 to 1250 to Chapters 148-162, from
a.d. 1250 to 1268 to Chapter 163, and from a.d. 1268 to 1300 to Chapters
164-167.
At the end of the list, we made use of the fact that Chapters 141-167 du-
plicate Chapters 103-137. Thus, we introduced the same time scale in both
texts X and B. After having calculated the volumes of fragments which
described each of the 19 intervals, they were averaged, i.e., divided by the
length (in terms of generations) of the interval. For example, the volume of
Chapters 2-14 describing Interval 1 equals 59 verses, whereas the length of
the interval spans 13 generations, and the average value of the volume per
one generation equals 59 -r 13 = 4.54 (see the graphs in Fig. 33, top). All the
local maxima (peaks) of both volume graphs are indicated in black. It is ex-
tremely surprising that all of them, except one, occur at the same points. It
is also important that all epoch duplicates of the series T, designated by tri-
angles in Fig. 33 (bottom), almost coincide with the peaks of the unaveraged
volume graph constructed for the biblical Chapters which were determined
for Generations 1-137. It can be seen that all the triangles are placed close
to the basic peaks of the volume graphs. In particular, all these duplicate
epochs of the series T immediately stand out from the entire quantity of the
biblical chapter generations at least because their volume graph exhibits lo-
cal jumps, or peaks. Duplicates of the form T are those described best of all
among the Chapters on line B. Following the method of [2] for calculating
§13
Construction of the Global Chronological Diagram
171
Old Testament: averaged
volume function 994 295
Roman history: averaged
volume function
j£9 99 9 5 5 * 8 8 9 B 8 i 88 ||
T T T TT T TTT T
Figs. 33-34. Top: Correlation between the volume function for Roman history
and the volume function for biblical history ( Old Testament). Bottom: Original
(unaveraged) volume function for the Old Testament.
p(X f Y ), we can estimate the nearness of these two series of peaks for both
graphs quantitatively.
Let us calculate the lengths of the intervals into which they break the
sequence of Chapter-Generation groups 1,2,... ,19. We obtain d(X,Y) =
1.4 • 10' 4 , which indicates that the texts X and B are dependent for eight
maxima.
This proximity is sq close that it is the least possible for two noncoinci-
dent vectors in the discrete model (see [2]), since they diverge in only one
172
New Experimental and Statistical Methods
Chapter 3
coordinate. Hence, within the framework of the maximum correlation princi-
ple, X and B describe the “same” events, which certainly sharply contradicts
today's view of the contents and relative historical epochs.
13.6.2. The overlapping of some biblical and European events. Baby-
lonian captivity and Avignon exile. Overlapping of the described his-
torical events, occurring in X and the Old Testament, implies, in particular,
that of the kingdom of Israel and Judah, described in the First and Second
Book of Samuel and Kings, the First and Second Book of the Chronicles,
coincides with part of the Holy Roman Empire in 962-1300, which is ideally
consistent with the independent overlapping obtained above on the basis of
the independent duplicate dynasty recognition method. These dynasties over-
lap due to the anomalous smallness of the coefficient A (A/, H ), which indicates
a dependence of the dynasties.
Let me stress once again that all the chronological results obtained by the
described methods are perfectly consistent with each other, which is a serious
argument for the objectivity of the duplicate system discovered.
Overlapping of biblical and European (in particular, Italian and German)
events leads, inter alia , to the following identifications. The famous events
under King Zedekiah (war with Pharaoh and Nebuchadnezzar, the fall of the
kingdom of Judah, capture of Jerusalem and Babylonian captivity) overlap
with those at the end of the 13th century in Italy, namely, the war in Italy,
capturing of Rome, transfer of the pontificate to Avignon, and complete sub-
ordination of the papacy to the French crown (“exile of the popes”). The
biblical 70-year-long Babylonian captivity is a reflection (duplicate) of the
70-year-long Avignon exile of the popes in 1305-1376 [6].
The medieval authors in the 14— 15th century confirm our conclusion, calling
the Avignon exile of the popes “Babylonian captivity” (see, e.g., [81] p. 112).
In particular, Dante wrote in a letter to the Roman king Henry, dated April
16, 1311, discussing the Avignon exile of the popes, that the heritage, which,
being deprived of it, they continued to be feel sorry for, will be returned to
them fully. And, similarly to their yearning for holy Jerusalem (Rome?), the
Babylonian exiles (in Avignon!) having become citizens, will joyfully remem-
ber the suffering of the turbulent years.
The further biblical events described in Ezra, Nehemiah, and Esther (return
to Jerusalem, “restoration of the temple”) are reflections of the corresponding
events in Italy in 1376-1410 (return of the pontificate to Rome).
For the convenience of comparing the biblical and European events, we
decode the letter symbols of line B (Bible), indicating the plot of the corre-
sponding biblical legend for each of them. Thus, B =
T : legend of Adam and Eve;
K: Cain and Abel, Enoch, Irad, Methushael, Lamech, Seth, Enosh, Canaan,
Mahalalel, Jared;
T : Noah, the Flood, destruction and rebirth of mankind;
H : Shem, Ham, Japheth, “sons of Japheth;”
T : Babylonian captivity, dispersing mankind all over the world;
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Construction of the Global Chronological Diagram
173
K : Arphaxad, Shelah, Eber, Peleg, Reu, Serug, Naher, Ferah, Abram;
T : Abraham and Sarah, fighting the Pharaoh;
K : Abraham, Haran, separation into two kingdoms, major biblical patri-
archs Isaac, Esau, Jacob, Judas, Joseph;
T: Joseph in Egypt, his serving the Pharaoh, the legend of the “woman”;
T : Moses, war with the Pharaoh, the Exodus, making the laws of the Is-
raelites;
H/U/P : Moses* death, Joshua, war and settling in the Promised Land,
story of the “Judges”;
T : sons of Benjamin, war; T : Ruth, Saul, Samuel, David;
K: kingdoms of Israel and Judah;
T : war with the Pharaoh and Nebuchadnezzar, fall of the kingdom of Judah,
beginning of Babylonian captivity (analogue of the Avignon exile), destruction
of Jerusalem;
C a : Babylonian captivity, return after 70 years, new “foundation” of the
temple.
To identify these events with the corresponding European ones, we have
to turn our attention to Fig. 32, in which line B is represented above, and
compare its symbols with the corresponding expansions of the “European”
symbols.
It follows from the decomposition of the GCD into the sum of four chroni-
cles that nearly all of the “modern textbook” , referring to dates earlier than
a.d. 900, consists of duplicates whose “originals” occurred in a.d. 900-1600.
In particular, each event described in the “modern textbook” before a.d. 900
is the sum of several (mainly, two, three, or four) later events. To find the
authentic dates when these events did occur, we have to draw a vertical line
and mark those events which it cuts out of the four chronicle lines Ci, C 2 ,
C 3 , and Ca •
In other words, the “modern textbook” is a fibred chronicle glued together
from the four pieces shifted with respect to each other and being practically
identical. The GCD contains no duplicates, starting only with the middle of
the 13th century and later.
13.7. The beginning of “authentic” history in circa the
10th century A.D.
Duplicates may already be found in A.D. 900-1300, e.g., block C (see Fig. 32),
whose inverse image (preimage) is the Empire of the Habsburgs, and placed
above A.D. 1300. In particular, the part of the “modern textbook” from 900
to 1300 is the “sum” of two chronicles, namely, a certain authentic chronicle
describing actual events in 900-1300 (probably, rather poorly), and another
authentic chronicle shifted backwards by ca. 300 years and describing the
events of the time of the Habsburgs, 1300-1600.
Global chronology was created at the end of the 16th century or at the
beginning of the 17th century, where the last period C “dropping” backwards
due to chronological errors and generating duplicates “in antiquity” (see the
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New Experimental and Statistical Methods
Chapter 3
letters C in the GCD) ends just here, which is very important and means
that no events subsequent to the times of Scaliger and Petavius were shifted
backwards. On the other hand, many events up to this epoch were dropped
backwards. It is probable that the last event dropping backwards was the
activity of the creator of chronology Petavius, or Petit. Translated from the
French, it means “little”, i.e., Dionysius Petavius is Dionysius the Little, the
famous chronologist who lived in the 6th century A.D. It is probably he who
may be called the “reflection” of Dionysius Petavius in his being dropped
backwards by ca. 1,000 years.
The whole of the GCD is a fibred document; many events regarded as
ancient are the sums of several later events described in the chronicles C2,
C3, and C4 glued to C\. The application of our methods to A.D. 1300-1900
did not lead to the discovery of any duplicates at all, which demonstrates
the validity of the historical scheme from 1300 to 1900. The “textbook” GCD
arose from the shorter chronicle C\ (or C 0 ) not due to some global falsification,
but, probably, due to three simple chronological errors, one of which was
the confusion between the dates of the foundation of two Romes, namely, in
Italy and on the Bosphorus (Constantinople = New Rome). The resolution
of the GCD into the sum of four chronicles supplies the answer to the two
fundamental problems, namely, what “authentic” history was, and how the
“modern textbook” was obtained from it.
Apparently, “authentic” history starts with ca. the 10th or 11th century
A.D. (or even later); before the 10th century, there are quite insignificant data
referring to A.D. 300-1000. All the other epochs mentioned in the “modern
textbook” as dated to earlier than the 10th century A.D. are various reflections
of the events in the 10— 17th century. Biblical New and Old Testament history
is placed in the interval from the 10th to the 15th century.
13.8. The chronological version of Morozov and the author’s
conception
The author’s conception is different from the version of Morozov as much as
his theory differs from the traditional one. For example, according to Moro-
zov, the basic biblical events occurred in the 3rd-5th century, which is ap-
proximately 1,000 years later than the traditional dating, whereas, due to the
results obtained by our methods, they occurred in the 10— 15th century, i.e.,
ca. a millennium later.
My conception given here completely rejects the hypothesis formulated in
[1] that most of ancient texts are allegedly Renaissance fabrications. As seen
from the GCD, all those written just before and during the Renaissance de-
scribe the authentic contemporary events and are in no way fabrications; for
example, Ptolemy’s Almagest. Morozov’s main accusation that the book had
been forged was that it had been speaking of the astronomical observations un-
der the Roman emperor Antonius Pius, whereas the authentic data (we mean
here the Latin text; see above) explicitly point to the 16th century. However,
there is no contradiction. The Roman emperor Antoninus Pius turns out to be
§13
Construction of the Global Chronological Diagram
175
placed in 1524-1547 (whereas dated traditionally, his rule was in 138-161 [6])
under the total forward GCD shift by 1,053 + 333 = 1,386 years (see Fig. 32).
It is surprising that "Antoninus Pius” is placed exactly in the epoch of the
first editions of the Almagest , the Latin one in 1537, the Greek one in 1538,
and the "translation” of George of Trebizon in 1528, published under "Anton-
inus Pius” , who was mentioned in the text. The author of the Latin edition
deceived no one, writing into the book the name of the ruler at the time of
the observations. We have a remarkable opportunity to verify the result by
another independent technique. With the Roman Empire in the lst-3rd cen-
tury being overlaid on the 10-13th century and the Habsburgs (see the GCD),
we can attempt to directly indicate a Habsburg whose name was "Pius” . The
epoch just before the first edition of the Almagest , the beginning of the 16th
century, is "covered” by Emperor Maximilian I (1440-1519). At least part of
the astronomical observations had to be made just in his time if the edition of
the book followed immediately after it had been written. The complete name
of this emperor turns out to contain the names Kaiser, Pius, and Augustus
(Fig. 9(2)). The epochs of Alberti and Vitruvius ideally coincide upon shifting
the GCD in the indicated manner [1].
We now describe the mechanism of a possible error which led to the "modern
textbook” being obtained from the shorter chronicle C Q -
13.9. The confusion between the two Romes
The first chronologist (possibly, in the 13th or 14th century) possessed several
documents of approximately the same contents, describing the same Roman
history, e.g., several versions of the type of Livy’s History of Rome. Written by
different people and from different points of view, in different languages, with
the use of different names or nicknames for the same historical figures (kings),
these chronicles were outwardly sharply different. The natural problem of re-
lating these documents to each other arose; in particular, the chronologists
faced the problem on which basis such a relation should rest. One of the
suggested methods was, probably, as follows. In many chronicles, years were
counted from the "foundation of the City (Rome)”; see, e.g., Livy’s History
of Rome. Therefore, to relate documents of this type to medieval chronicles,
it sufficed to calculate the date of the "foundation”. However, many medieval
documents confuse the two Romes, namely, in Italy and on the Bosphorus. It
is assumed that Constantine I transferred the capital in ca. 330 A.D. from the
Italian Rome to the settlement Byzantium on the Bosphorus, which was offi-
cially named "New Rome” [76]. It is only later that New Rome was called Con-
stantinople [76]. Both Romes were the capitals of great empires. It had been
stressed long ago that the citizens of New Rome stubbornly called themselves
Romans (they were allegedly called the Romaics by other people); hence, the
Romaic Empire is the Roman Empire (as well as the Italian one). Along with
the legend about the capital’s transfer from the Italian Rome to the Rome on
the Bosphorus, a similar legend speaks about transferring the Empire’s capital
176 New Experimental and Statistical Methods Chapter 3
from the Bosphorus to Italy. This attempt was allegedly made in A.D. 663 and
again by emperor Constantine (but now Constantine III and not I), who did
not complete the enterprise, because he was murdered in Italy [1], It is gener-
ally assumed that the Rome on the Bosphorus was the Greek capital. However,
a large percentage of Byzantine coins had Latin, not Greek inscriptions (so
did Italian coins). As the famous legend about the foundation of Rome has
it, two cities were indeed founded: one by Romulus, and the other by Remus
(see Livy). Both founders have similar names; then Romulus “killed” Remus,
and only one Rome remained, i.e., the capital (Livy, Bk. 1, 1), which possibly
reflects the confusion between the two Romes; so much so that certain ancient
chronicles called the founders of both capitals Romulus and Rome, but not
Romulus and Remus, which almost identifies the founders’ names [51]. See
also the Russian edition, 1911, p. 18.1, B. 170-175.
It is assumed today that it is always the Rome in Italy that is under-
stood to be the “City” from whose foundation years were counted in Roman
documents. But the medieval authors in the 12-14th century were not so
categorical. Moreover, in the words of Villehardouin, 5
“. . . the city which was sovereign over all the others, and which the Byzantines
willingly called simply ‘the City* (! — A.F.), i.e., the city par excellence, the unique
city” ([93], p. 14).
Thus, counting years from the foundation of the City may imply Rome
on the Bosphorus in many a document. It is assumed that Constantine I
transferred many institutions from Rome to Constantinople and ordered the
construction of palaces precisely copying the senators’ Roman homes. The
Byzantine Empire continued to be called the Roman Empire [93].
On the contrary, the backward “influence” of New Rome on the Rome in
Italy was great:
“. . . 7-13th-century Rome was a semi- Byzantine city . . . the Greek customs were
practiced everywhere; the Greek was the official and even habitual language of the
country . . . they continued to resort to Greek for the official acts as well as for cor-
rect usage The Norman kings were proud to bear the magnificent costumes of
Byzantine emperors ...” ([93], p. 19).
The so-called fiction onto which the Byzantines stubbornly held for cen-
turies, claiming that they were authentic Romans, has repeatedly been noted
in traditional history. The Byzantine emperors continued to look upon them-
selves as the only legitimate Roman emperors. With all the Byzantine his-
torians, all the Greeks turned out to be “Romans”. Fearing ambiguity, they
arbitrarily [!? — A.F.] called the Byzantine Empire Romaic. The name Ro-
mania was transferred from the Byzantine Empire to the Ravenna exarchate
[77].
It is not accidental that we have discussed the confusion between the two
Romes in such a detailed manner. Assume now that an ancient chronicler has
5 Translated from the French — tr.
§13
Construction of the Global Chronological Diagram
177
made a natural mistake (“without malice”) and mixed up the “foundation”
of New Rome in A.D. 663 with that in 330 (see above). Then, by making the
foundation of Rome in A.D. 330 on the chronicle C\ (see the GCD) coincident
with that in A.D. 663 on C 2 , he let C 2 drop 333 years backwards with respect to
Ci (because 333 = 663—330). Then, by patching these two chronicles together,
he obtained a “longer” history with duplicates. That was, probably, the reason
for the first shift back by 333 years, which led to the creation of “authentic”
history. Similar reasons can be indicated also for the other two shifts by 1,053
and 1,778 years. However, we should note that the latter two shifts can, in
reality, be generated by the 333-year one. As a matter of fact, certain chronicles
in the GCD turn out to have been shifted not by 333, but by 360 years (a
difference in 27 years equals the duration of the war (T), the most frequent
duplicate in the GCD). The shift by 1,053 years can then be represented as
1053 = 360 + 360 + 333, whereas 1053 + 720 = 720 + 720 + 333 = 1773, which
all but coincides with the 1,778-year shift. Thus, it is possible that the three
basic shifts are multiple ones, i.e., repetitions, of the first one by 333 (or 360)
years.
13.10. A universal mechanism which could lead to the chroniclers 9
chronological errors
The author’s paper New Empirico-Statistical Methods of Ordering Texts and
Applications to Dating Problems [97] offers the following, possibly univer-
sal, mechanism which could lead to the chroniclers’ errors and result in the
three basic chronological shifts. Probably, the primary basic dates from which
the chroniclers started counting years were written with the literal symbols
making up a short verbal formula. The original meaning was soon forgotten.
Afterwards, the chronologists decoded the old dates by formally replacing the
letters by figures (it is generally known that figures mean letters in ancient
languages), following the standard rules, namely, A = 1, B = 2, etc., and
could obtain totally different results.
For example, the abbreviation “XHIth century A.D”. could originally have
meant X.III, i.e., the “3rd century since Christ”, where X is the well-known
anagram of Christ. Substituting formally, we obtain a 1,000-year backward
shift. Similarly, A.D. 1500 = 1.500, i.e., the “500th year since Jesus”. Mean-
while, the original count was probably made since the times of Gregory VII
Hildebrand (11th century A.D.). Consider two other important dates related
to him, namely, 1073, the year of his election as pope, and 1075, Cencius’
conspiracy and the lunar eclipse at the Crucifixion. Recalculating and trans-
lating them into those since the Byzantine creation of the world, we obtain
1073 + 5508 = 6581, and 1075 4* 5508 = 6583. Writing the figures in letters
according to the usual rules, we obtain 6581 =^S$nA, 6583 =^S$nr. The
sign =f= is only regarded today as the sign for “one thousand”. However, it
may have been a distorted form of the letter “I”, i.e., an abbreviation of the
word “Jesus”. $ was written in the same manner as 0, and we obtain two
quite meaningful word dates, namely, 6581 = IS0nA, 6583 = IS0ni\ Indeed,
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New Experimental and Statistical Methods
Chapter 3
IS0IIA = Jesus God Pope Augustus, and ISQIir = Jesus God Pope Gregory
(or Hildebrand), where IS = Jesus, Qeov = God, n = Pope, and A = Augus-
tus. Repeating the procedure in reverse order, we discover the reason which
makes the chronologists believe that the “creation of the world” occurred in
5508 B.c. They just did not recognize the abbreviations of certain important
word symbols.
A similar mechanism might possibly lay the foundation for the shift of 333
or 360 years. The dates from the end of the 15th century to the beginning of
the 16th century, i.e., during the rule of Emperor Maximilian I (1493-1519),
could be written by the chronologists, for example, as follows: MCL.III, which
originally meant the “3rd year since Maximilian” , where M.C.L. might be spelt
out as Maximus Caesar Leo, or Great Kaiser Leo. During a later substitution
of figures for Latin letters, an erroneous “date” was obtained, namely, 1153,
which differed by 343 years from the authentic one, since 1493 = 1493+3, and
1496—1153 = 343. Thus, the documents using the abbreviated formula M.C.L.
at their later decoding were automatically shifted backwards by ca. 340 years.
A similar natural “verbal” formula is also the basis for the third chronological
shift by ca. 1,800 years. We stress that the letters in the dates are, in fact,
separated from each other by periods in certain ancient documents, e.g., on
Diirer’s prints.
After all the ancient chronicles have been “returned home” and placed prop-
erly in the 10th-17th century, we obtain that the history of Europe, the Near
East, and Egypt is known approximately as much as that of the so-called
“young cultures” such as Scandinavia, Russia, or Japan. It is probable that
this cultural “alignment” reflects the natural fact that existing civilizations
and developing ones were born more or less simultaneously in different regions.
13.11. Scaliger, Petavius, and the Council of Trent. Creation of
traditional chronology
We have noted above that the GCD duplicates were discovered only for periods
preceding Scaliger’s epoch, but not for later ones. Thus, we again see that the
times of Scaliger and Petavius are somehow related to the discovered effects
in ancient chronology and history. Recall that it was the group of Scaliger and
Petavius which had fixed “historical tradition” which laid the foundation of
the “modern textbook”, the GCD. Scaliger’s version happened to be created
within the animated chronological controversy at the end of the 16th century
and the beginning of the 17th century. Moreover, Scaliger’s version turned out
not to be unique at all. It confronted some other points of view whose partisans
have lost the fight. For example, here are some facts regarding certain events
of those turbulent times, the epoch of the 30-year European war, chaos and
anarchy.
“It suffices to recall the famous chronologist Joseph Scaliger opposing the Gregorian
reform” ([78], p. 99).
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Construction of the Global Chronological Diagram
179
Its preparation started in A.D. 1514 during the Lateran council. It is as-
sumed today that the principal problem related to the reform was a shift of
the equinox. But it is only one of many serious issues debated in connection
with the calendar revision. One of the items leading to the greatest controversy
was Scaliger’s so-called Julian period. The “great” indiction is a 532-year pe-
riod, which was called indiction (as believed today) in the Byzantine Empire,
and the great circle in the West.
“It is difficult to determine with sufficient accuracy when and where this period
first came into use” [78].
It is believed (the originals of the documents having been lost) that it was
known to the paschalists of the council of Nicaea in the 6th century A.D. There
also exists a modification of this “Great Indiction”, namely, a period of 7,980
years, also regarded as “ancient”. However, as suddenly becomes clear,
. . it happened so that this ancient cycle was accepted by the science of chronology
only at the end of the 16th century A.D., and then ... said to be * Julian*. It was
introduced into science by the outstanding scientist and erudite J. Scaliger (1540-
1609) in his Opus novum de emendations iemporum This work saw the light in
1583, almost simultaneously (! — A.F.) with the Gregorian reform whose adversary
of principle the scientist remained until his death. (Here, the creation of a global-
chronology ancient- world calendar is already meant — A.F.) Referring to the Byzantine
works, Scaliger insisted on the Julian calendar as the only chronological system being
capable of supplying a continuous year count in world chronology” ([78], p. 106).
Controversy surrounded the chronology and the whole of Scaliger’s concep-
tion:
“It is paradoxical in this sense that the very period (of Scaliger — A.F.) indispensable
for . . . the chronology of our times was looked upon by pope Gregory XIII as unsuitable
for calendar purposes” ([78], p. 107)
The famous Council of Trent (1545-1563) took place at the same time.
Among other things, the biblical Canon and the famous chronology “since
the creation of the world” were fixed just then. In general, the whole epoch is
characterized by the struggle with Protestantism.
“The central tribunal of the Inquisition was created in Rome . . . and the index
of banned books issued. . . . The Council of Tent played an important part in these
reactionary measures taken by the Catholic Church. . . . All Protestant works and
their teachings were anathemized The importance of the Council of Trent for the
subsequent activities of the Catholic Church was extreme” ([79], Vol. 2, pp. 107-108).
Scaliger’s chronological work, playing an important role in substantiating
the authority and old age of the institutes of the Catholic Church, which grew
out of Roman history, was published just at that time. In my opinion, it is
necessary to study the archives of the Council of Trent and to revise all the
surviving documents of those turbulent years, which might cast light on the
controversy surrounding Scaliger’s chronology.
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Chapter 3
§14. The “Dark Ages” in Medieval History
14.1. Medieval Italy and Rome
As can be seen from the GCD and its decomposition into the sum of three
shifts, almost all documents regarded today as ancient, and describing the
events traditionally dated to earlier than A.D. 900, probably duplicate the
originals which supplied the account of the 10-17th-century events. The ques-
tion arises whether the “ancient world” can be placed in medieval history,
i.e., whether we shall not find the position of the allegedly old texts in the
Middle Ages, since they are closely packed with the medieval events already
known to us. But, as detailed analysis shows, this is not so. First, certain
epochs earlier regarded as different should be identified (for example, we need
to overlap several dynasties on one another; see Section 4). The similarity
of these duplicates has previously been unobserved. Second, many periods of
medieval history are in the dark due to the complete (or partial) absence of
the corresponding documents “shifted backwards”. Their removal and shift-
ing into “ancient times” have immersed many periods of the Middle Ages into
artificial darkness. In the 18-19th century, a peculiar point of view spread
among the historians that the Middle Ages had been “dark ages” .
“The great achievements of antiquity were allegedly forgotten, scientific thought
descended to the ‘cavemen’s level’, great literary works of the ancient times lay as a
dead weight, and became known only during the Renaissance” ([80], p. 161).
Most of the upper clergy were allegedly illiterate ([80], p. 166). Coin minting
stopped, architectural art fell into oblivion, culture everywhere was “running
wild”, and so on ([80], p. 167). In our opinion, we face not the degradation
of the “great heritage of the past”, but the birth of civilization gradually
creating all those cultural and historical treasures partly dated earlier in the
past due to certain chronological errors, mistakingly leading to the illusion of
enlightened “antiquity” , and leaving some periods of the Middle Ages bare. For
example, the existing medieval history of Rome reveals an enormous quantity
of obscure passages, contradictions, and obvious absurdities if we consider it
at close range, and which can be explained by the distorted chronological idea
of the role of the Middle Ages.
Due to the leading role of Roman chronology (see above), we now describe
in a nutshell the situation of the history of Rome.
“With the overthrow of the Gothic kingdom begins the ruin of the Italy and Rome
of antiquity. The laws, the monuments, even the historic recollections of the past fade
from memory” ([53], Vol. 2, p. 1).
The removal of secular chronicles (e.g., Livy’s History) from Roman me-
dieval history has turned Rome into a wholly spiritual city if we look at it
from the modern standpoint:
“The metropolis of the universe was converted into a spiritual city” ([53], Vol. 3,
p. 3).
F. Gregorovius says that this transformation of the “civil Rome” into the
§14 The “Dark Ages * in Medieval History 181
“religious City” was declared a great and remarkable metamorphosis in the
history of mankind.
Speaking of the end of the 6th century A.D., Gregorovius, the author of the
most fundamental work [53], declares:
“The following years are hid in obscurity. The chronicles of the time, monosyllabic
and dismal as itself, speak of nothing but the havoc ...” ([53], Vol. 2, P. 1, p. 24).
He continues about the middle of the 9th century:
“The papal archives contained the innumerable acts of the church and the regesta”
([53], Vol. 3, P. 1, p. 141).
And then all these documents were lost in the 12th-13th centuries, leading
to the large gap in our knowledge about this epoch.
“Did we but possess these regesta, the history of Rome from the seventh to the
tenth century would live anew” ([53], Vol. 3, p. 141, note).
“The history of the city, its remarkable transformation since the days of Pipin and
Charles, found not a single analyst, and while Germany and FVance, and even Southern
Italy . . . produced numerous chronicles, the indolence of Roman monks allowed the
events of the city to remain shrouded in profound obscurity” ([53], Vol. 3, p. 147).
From time to time, the medieval chronicles report about “ancient” facts
as of contemporary ones. The historians then start speaking of resurrected
remembrance, reminiscences, imitation of the old customs, and so forth. For
example,
“. . . the Romans of the tenth century are frequently designated by curious-sounding
names. These names arrest our attention, recalling as they do the monuments of
antiquity” ([53], Vol. 3, p. 381).
The discussion of the problem regarding the existence of a senate and con-
sulship in medieval Rome has burst out in traditional history many times.
Some believe that all these institutions (regarded as “ancient”) also existed in
the Middle Ages, whereas the others declare that medieval Romans followed
these “ancient customs” under their own momentum, not giving them their
earlier meaning.
Gregorovius wrote:
“. . . the aristocrats, the citizens, the militia . . . summoned to their aid from the
already myth-enshrouded graves of antiquity the ghosts of consuls, tribunes, and sen-
ators, who seem to have haunted Rome throughout the entire Middle Ages ” ([53],
Vol. 2, p. 417).
“. . . the title consul is very frequent in documents of the 10th century” ([53], Vol. 3,
p. 450, note).
And moreover, Emperor Otto (in the 10th century) tried to recover “an-
cient” and “forgotten” Roman customs. Speaking of the description of me-
dieval Rome, which was given in the famous medieval book Graphta , Gre-
gorovius shamefacedly declares that the Graphia mixes antiquity and medieval
reality:
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“The Graphia still calls it Templum . . . and relates the legend of the earth being
piled up” ([53], Vol. 3).
Medieval chronicles very often speak of the facts which are contrary to
traditional chronology; this confirms the three shifts we discovered in the
GCD.
Thus, it turns out that
“Noah founded ... a city near Rome, which bore the name of the founder . . .
his sons Janus, Japhet, and Camese built the town of Janiculum on the Palatine. . . .
Janus dwelt on the Palatine, and aided by Nimrod . . . built the city of Satumia on
the Capitol” ([53], Vol. 3, p. 526).
“In the Middle Ages a monument in the Forum of Nerva was called Noah's Ark”
([53], Vol. 3, p. 527, note).
All these "absurd” things (from the standpoint of tradition) precisely cor-
respond to the overlap of the kingdoms of Israel and Judah and the Empire
of the 10-13th century A.D. in Italy.
In general,
“And it is solely by means of this antique character — a character which dominated
the city throughout the entire Middle Ages — that many historic phenomena can be
explained” ([53], Vol. 3, p. 537).
It turns out that the first lists of Roman monuments were made only in the
12th century and are assumed today to be w . . . a curious medley of true and
false names” ([53], Vol. 3, p. 543). For example,
“The church was dedicated not only to S. Sergius, but also to S. Bacchus, a saint,
who, curiously enough, appears on this ancient Pagan site. His appearance, however,
was not singular in Rome, where the names of ancient gods and heroes are again found
among Christian saints, as S. Achilleus, S. Quirinus, Dionysius, Hippolytus, Hermes
...” ([53], Vol. 3, p. 544).
The history of the world-known architectural monuments of Rome can be
more or less reliably traced to not earlier than the 10th or 13th century A.D.
For example,
. . for a long time past (after antiquity — A.F.) we have not once heard its name
(Capitol). It had vanished from history (at that time, it had not even been built —
A.F.)” ([53], Vol. 3, p. 546).
The chaos leading to the complete confusion among "ancient” and "me-
dieval” names reigns over the medieval names of Roman monuments. For
example,
“The Temple of Vesta was formerly made into the Temple of a Hercules Victor; the
archaeologists have now dedicated it to Cybele; this Goddess will, however (? — A.F.),
soon have to withdraw to make room for another divinity, until the latter is in turn
banished by an archaeological revolution” ([53], Vol. 3, p. 561).
This more resembles a game than a science.
“Night, however, veils the most exalted spot in history (the Capitol and its suburbs —
A.F.) for more than five hundred years. ... It was merely the inextinguishable tradition
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183
of all that the Capitol had once signified which now raised it from obscurity, and which,
as soon as the spirit of civic freedom was awakened, made it once more (! — A.F.) the
political head of the city. As early as the eleventh century the Capitol appears as the
centre of all purely civic affairs (among the ruins? Traditional history assures us that
the Capitol was destroyed as early as the most ancient past, and has been standing in
this practically erased form until the present day — A.F.). The recollection of the sacred
spot was revived; the ruins of the Capitol (! — A.F.) reanimated by the assemblies of
the nobles and people now usurped the place of the Tria Fata ... at the disturbances
on the election of a prefect, at the acclamation of the election of Calixtus II, it was
again from the Capitol that the Romans were summoned to parliament or to arms. It
would also appear that the city prefect dwelt on ‘the Capitol, since the prefect, Henry
IV . . . , had a seat there . . . and a palace on the hill was used for tribunals . . . (also
among the ruins? — A.F.) ...” ([53], Vol. 4, P. 1, pp. 464-465).
Can we accept, even as a hypothesis, that all these assemblies, meetings,
elections, arguments, discussions of documents (and keeping them), making
responsible state decisions, signing official papers, and so on, had taken place
in overgrown ruins and not in special buildings erected just for this purpose,
and just at this time, whereas they were destroyed much later, since there
were enough “waves of destruction” in ll-13th-century Rome. The mist of
orthodox conception veils Gregorovius so densely (he being one of the most
serious historians of Rome and the Middle Ages, whose works are always well
documented) that he continues his story, apparently not feeling the absurdity
of the described picture, which contradicts elementary common sense:
“Sitting on the prostrate columns of the Temple of Jupiter, or within the vaults of
the office of the State Archives, among mutilated statues and inscriptions, the monk
of the Capitol, the rapacious Consul, or the ignorant Senator might gaze in wonder at
the ruins and meditate on the capriciousness of fortune ...” ([53], Vol. 4, p. 465).
Without noticing the comical improbability of such legislative assemblies
under popes claiming world supremacy, Gregorovius continues:
“The Senators who went to and fro among the ruins, wearing tall mitres and gold-
brocaded mantles, had but a dim idea that here in former time Statesmen had framed
laws, orators had made speeches — There is no more bitter satire on all the most
exalted things on Earth . . . goat herds already clambered over the marble ruins (and
among the Senators sitting on them — A.F.); a part of the Capitol had even received
the degraded name of Goat-hill (Monte Caprino) in the same way that the Forum had
been transformed into the Campo Vaccino (not for the Senators? — A.F.) ...” ([53],
Vol. 4, P. 2, p. 467).
In confirmation, Gregorovius adds the medieval description of the Capi-
tol, the only primary source up to the 12th century. It is most striking that
this text occupying a whole page of a large-format modern book, printed in
brevier, reports no destruction and describes the medieval Capitol as a func-
tioning political centre of medieval Rome. Magnificent buildings, temples, and
so forth, are mentioned, but no word is spoken about the goat herds lonely
wandering among this golden luxury.
In the Middle Ages, the Basilica Constantini was called the temple of Ro-
mulus (!). Ricobald asserted that the famous “ancient” equestrian statue of
Marcus Aurelius had been cast and erected by the order of pope Clement III
184 New Experimental and Statistical Methods Chapter 3
(and it all was in the 12th century A.D.) ([53], Vol. 4, p. 586, Comm. 74).
Gregorovius dejectedly comments:
“This is the erroneous statement of Ricobald” ([53], Vol. 4, p. 698, Note 2).
The argument is that a similar work of bronze could not have been made
in the Rome of the time, with its very low level of artistic development ([53],
Vol. 4, P. 2, p. 666).
The medieval chronicles very often speak of the ancient way of contempo-
rary Roman life. For example, about the 13th century, it is said:
“As in ancient times, in the days of Camillus and Coriolanus, they (the Roman
people — A.F.) undertook conquering expeditions against Tuscany and Latium. The
Roman insignia, the ancient initials S.P.Q.R. ... were seen once more in the field”
([53], Vol. 5, P. 1, p. 164).
The important question arises: What was the Christian cult before Gregory
Hildebrand (11th century a.d.)? The study of this problem demonstrates that
it coincided with the “ancient” bacchanalian cult. Traditional history retained
many traces of this Christian bacchanalian religious service. For example, the
medieval papacy and clergy are believed today to have sunk into perversion
(famous “agapes”, nights of love, which were devoted not to friendly boozing
together, but to bacchanalian orgies). Certainly, it was not simple to abol-
ish the bacchanalian cult (due to its attractiveness); Hildebrand gave many
years of his life to the purpose. Afterwards, the Inquisition was summoned.
The famous medieval descriptions of the “witches’ sabbath” imitate the same
“agapae” now turned into a “devilish plot” (from the point of view of the
Church reformers after Hildebrand). It is natural that the new, Reformation
Church, which had originated in the 11th century under Hildebrand, shifted
the responsibility for the bacchanalia to the devil in order to stifle in the
flock the recollection of its own quite recent cult. In spite of the success of
Hildebrand’s reforms, the Christian bacchanal cult was still in force for a long
time in Western Europe. Here is, for example, Champfleury’s Histoire de la
Caricature an Moyen Age (see [1] for its analysis). Normally, cartoons resort
to certain real features in order to exaggerate them.
“Singular rejoicings took place in the cathedrals and convents apropos the great
feasts of the Church during the Middle Ages and Renaissance. At Easter, and especially
Christmas, it was not only the low clergy that took part in songs and dances, but the
great dignitaries of the Church. In the cloisters, the monks danced with the nuns of
neighbouring convents; the bishops went to look for the religious women to engage
themselves ii\ their joy” ([101], p. 53). 6
Champfleury then presents a picture of the monks’ meal and their lovers
from a Bible (!) of the 14th century, which is kept in the Bibliotheque Imperiale
in Paris (n° 166), as the most modest example of a comic cartoon! But how
can a “comic picture” — if it is, in fact, a caricature — get into the Bible, a holy
6 Translated from the French — tr.
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The a Dark Ages * in Medieval History
185
book? Holy texts have not been made for jokes or laughs; the more so that the
other miniatures of this edition do not at all manifest a joker in the illustrator.
The miniature presents a typical bacchanalian situation, namely, one of the
monks in front is engaged in amorous escapades with a nun, and the same is
repeated in the background, but now by a number of monks. There were a
great many “caricatures” in the medieval texts and Bibles. By the way, pope
Pius II, for example, was the author of a number of pieces of erotic literature
and an extremely indecent (from the modern standpoint) comedy Christ ([81],
p. 156). We also have to mention the famous Song of Songs included in the
biblical Canon, which is pervaded with transparent eroticism (treated as a
certain “allegory” by the modern theologians).
Trying to accommodate the life of the medieval monks to modern morality
and our modern ideas of the religious life of that time, Champfleury assures
us that all these pictures and texts should be regarded not as illustrations of
something that has really occurred, but as a warning against similar deeds. It
sounds strange, because the “warning” is pictured in a very attractive man-
ner. For example, who would warn people against perversion by circulating
magnificently illustrated pornography? Most probably, this would have the
opposite effect. Moreover, if the warnings had been in earnest, then some par-
ticular unpleasant consequences of such practice would have been represented.
But there was nothing of the kind there! Similar illustrations (including bib-
lical ones) are only possible in the case where they picture the usual way of
life of the medieval priesthood, a fact which at the time of the Middle Ages
was regarded as normal by all; and if a painter had made it with the purpose
of condemning the customs which had already lost the approval of the new
ideology, then, as noted by Morozov, he would have represented the revels
in some hideous form, with devils carrying sinners to Hell, with atrocious
consequences of terrible diseases, and so on. Instead, many medieval Bibles
were illustrated not only with representations of bacchanalia, but also with
“antique pictures”, namely, vines being climbed by angels indistinguishable
from antique cupids, and so forth. I am referring to my personal familiarity
with old Bibles (e.g., in the Library of the Moscow Planetarium or Museum
of Rare Books at the Foreign Languages Library in Moscow).
The synod of Chalon-sur Saone forbade women to sing indecent songs
in church. Gregory of Tours protested against the monks’ masquerades at
Poitiers, which were extremely licentious.
Champfleury wrote: 7
t4 In 1212, the council of Paris forbade the nuns to celebrate the fete des fous. ‘From
feasts, where the phallus is accepted, everyone should abstain; still more, monks and
nuns are prohibited’” ([101], pp. 57-58).
The ban helped very little, since a Reformation bishop visiting the Roen
monasteries in 1245 reported that nuns in many numbers indulged in illegal
7 Translated from the FVench — tr.
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pleasures at the feasts. Such a bacchanal still continued in Besangon between
1284 and 1559. Charles VII again forbade this religious fete des fous at the
Cathedral of Troyes in 1430. The new evangelical papacy founded by Hilde-
brand was driving out the deep-rooted bacchanalian cult with extreme diffi-
culties (and for how long!).
“More than once I have looked at the cathedrals, searching for the secret of their
disturbing ornamentation, and each motif which I have taken to clarify my text seemed
taken from an unknown language.”
“What to think of a strange sculpture hidden on a pillar of the underground cathe-
dral of Bourges?” ([101]).
This sculpture represents a man’s buttocks made thoroughly and expres-
sively, and protruding from a column. It is placed in a spot convenient for love
games. How could the priests and believers who regularly came to this temple
for prayer tolerate such a sculpture when it had not yet been established as an
artifact left from times long past? Attempts to account for these effigies (and
there are many!) as “caricatures”, hewn in stone in holy temples, by those
who preach in them cannot be regarded as serious.
We quote again from Champfleury:
“It is possible to find an imagination paradoxical enough to determine the relation
of enormous facetiae (joke! — A.F.) with the place where they are displayed, and should
we not admit the Caprice which has not stopped the worker from the execution of a
similar detail?”
“On the walls of certain religious monuments is seen the representation of sexual
organs which are displayed complacently in the middle of religious details: echoes of
antique symbolism, these priapic symbols were sculptured with innocence by some
naive stonemason These ithyphallic remembrances of diverse cathedrals in the
centre of France are numerous in Gironde, and are characteristic of Bordeaux. Leo
Drouyn showed me curious specimens shamelessly displayed in the churches of his
province and which he hides in the bottom of his folders.
“Our excess of pudery deprives us of important knowledge. The silence which the
modem historians keep with respect to the symbolism of reproductive organs continues
to keep a veil over the attitude of those who want to establish the parallel between the
monuments of antiquity and those of the Middle Ages. Serious books on the cult of
the phallus, important drawings in its support, would vividly clarify the question and
would show the thinking of workers in the Middle Ages who were not embarrassed by
being reminded of ancient pagan cults” ([101], pp. 239-240).
All these effigies were in no way a humiliation of the Church and pur-
sued the same purpose of attracting new believers — before wide repression of
the old cult was waged by the Gregorian church — as the pictures of foaming
tankards on the doors of pubs. The famous “antique” pornographic images
discovered, for example, in Pompeii [47] are practically indistinguishable from
the Christian sculptures and pictures. And again “prudery” hinders wide sci-
entific circles from becoming familiar with a great number of effigies of the
sort. It turns out that
“. . . those of the pictures, which represent clearly erotic or indecent scenes so much
appreciated by the ancients (and also in the Middle Ages — A.F.) are kept under lock
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187
and key Somebody secretly . . . scraped off the obscene frescoes at night All
the pictures and effigies in Pompeii, incompatible with the modem ideas of decency,
have been recently placed in the secret department of the Bourbon museum” ([47],
p. 76).
Houses with stone phalli at their entrances were discovered in Pompeii ([47],
p. 120). The relation of phallic images with the Christian cult cannot be traced
exclusively to the European temples.
“In Egypt, too, phalli of monstrous size were hewn of granite They were placed
at the doors of the temple” ([47], p. 122).
V. Klassowski suggests that these giant stone effigies were placed there for
the “pilgrims’ edification” (?) ([47], p. 122).
The erotic sculptures of the Christian cult can be found on the capitols of
the cathedral in Magdeburg, on the walls of the Notre-Dame in Paris, finished
in the 12th century, and so on.
It is generally known from the archaeology of medieval Rome that prac-
tically each of the most important Roman Christian churches was allegedly
built on the “ruins” of earlier pagan temples, with these “pagan sanctuaries”
being erected approximately for the same purpose and even bearing the same
name as the Christian (and “later”) temples [53]. In my opinion, by declaring
its bacchanalian past (until the 11th century) “erroneous”, and by moving on
to a new evangelical phase (in the 10th-12th century), the Christian Church
simply renamed its prior pagan bacchanalian temples and declared the pagan
gods “new” evangelical saints.
Certain of the consequences of the three basic chronological shifts which I
discovered were stressed by different authors at different times. For example,
Gregorovius noted a certain parallel between the ancient and medieval events.
The following important statement is valid, namely, that all the instances of
the parallel , indicated by him , precisely correspond to the three chronological
shifts. We illustrate this by a simple example. Gregorovius directly points
to the parallel between the Gothic war in the 6th century A.D. and the war
in Italy in the 13th century, being perfectly right in identifying the parallel
personages.
“The gloomy Charles of Anjou appeared on the ancient battleground of the Ro-
mans and Germans like Narses (! — A.F.), while Manfred assumed the tragic aspect of
Totila (! — A.F.) ... for although the relations of powers were different, the conditions
remained essentially the same (! — A.F.). . . . The Swabian dynasty fell as that of the
Goths (! — A.F.) had once fallen. On one and the same classic stage the impressive
overthrow of two dominions and their heroes adorned history with a twofold tragedy,
of which the second seemed, as it were, to be merely the precise repetition of the
firstf!— A.F.)” ([53], Vol. 5, P. 2, p. 365).
14.2. Medieval Greece and Athens
The situation with the history of medieval Greece is considerably worse, in
the sense of completeness of information, than that of Rome. Like the his-
tory of other antique cities, that of Athens is characterized by blooming in
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ancient times followed by immersion into medieval darkness from which the
city started to emerge only halfway through the Middle Ages, and later than
Rome. Gregorovius wrote that, as to Athens proper, its medieval fate is cov-
ered by such impenetrable darkness that the most outrageous opinions have
been expressed about the city, such as claiming that Athens turned into an
uninhabited coppice in the 6-T0th century, and finally, was burnt out by the
Barbarians. Very valid proofs were supplied of the existence of Athens in the
gloomiest epoch. Let us note the remarkable fact that special proof should
be required only to prove at least the existence of the most glorious city in
the historical country. This fact serves as the startling confirmation of the
complete vanishing of Athens from the horizon of history [77].
These details about the situation in medieval Athens were first stated by
Fallmerayer in the 19th century. To somehow account for this enigmatic
“catastrophe”, he suggested that the Avars and Slavs massacred all of an-
cient Greece. However, no sources confirming this are available. Starting with
the 7th century A.D., Greece becomes so unimportant for history that the
names of Italian towns are much more often mentioned by the Byzantine
annalists than Corinth, Thebes, Sparta, or Athens. However, afterwards, too,
none of the chroniclers hinted with a single word at the capture or devastation
of Athens by invading peoples. Athens did become impoverished, its marine
power and political life fell into decay as well as the life of Hellas on the whole.
However, the fame of the medieval city was in the hands not as much of the
wise men as of honey-merchants. Athens and Hellas were covered by profound
night. The “ancient” Parthenon was strikingly turned into a Christian church!
The Virgin Mary had already started her victorious war with ancient Pallas
over the possession of Athens. The 10th-century Athenians built a beautiful
church and erected the image of the Mother of God, whom they called Athene
(! — A.F.). Moreover, giving the name of the Mother of God is the same as to
give the name of Athene the same name which was later given to the image
of Panaghia Athene highly revered in the medieval Pathenon. Thus, besides
the identity Athenae = Mother of God, we discover that the Parthenon was
devoted to the Mother of God, or Athene [77].
Gregorovius continues that the noblest of all cities on earth was hopelessly
immersed in the gloomiest Byzantine epoch. With ever-increasing contempt,
New Rome on the Bosphorus came to look down on the fallen guiding star
of Greece, the small provincial town of Athens. As to the fate of Athens’
monuments, they, as a rule, remained in oblivion. For hundreds of years, the
Greeks were sitting in total obscurity under the shelter of the ruins of their
grey-haired antiquity. Certain of the most beautiful ancient buildings lured the
Christians of Athens into rebuilding them into churches. We know nothing of
the day when this happened for the first time, and when a temple in Athens
was first turned into a Christian house of worship. The history of Athens’
churches is very uncertain. Speaking of the Parthenon, Gregorovius adds that
the Christian religion drew its attention to the great sacred spot, i.e., the
Parthenon, of the antique city goddess on the Acropolis, doing no harm to
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The u Dark Ages n in Medieval History
189
the temple at all. In the entire history of the transformation of ancient beliefs
and holy concepts into Christian ones, you will find no example of such an
easy and complete substitution as occurred to Athene Pallas when she was
replaced by the Virgin Mary. The people of Athens did not even have to alter
the nickname of their divine virginal protectress, for the Virgin Mary was now
called Parthenos by them [77].
Medieval Athens first appeared on the arena of history after many years of
nonexistence as a small Byzantine fortress allegedly “restored” by Justinian as
early as the 6th century A.D. in the territory fully populated by the Avars and
Slavs. No traces of “ancient” Greek Hellenes could as yet be found. In general,
according to Gregorovius, the whole Acropolis turned into a holy shrine of the
Virgin Mary. We do not possess any factual proofs of existence in Athens of
either schools or public libraries. The same darkness veils Athenian civic rule
in that epoch.
Why did the classical thought “evaporate” from Greece? Where did the
“classical” Greeks go? Why did the famous “ancient” marine potential of
Athens vanish (by the way, “restored” in the Crusaders’ epoch during the 12-
13th century)? The documents indicate that the Byzantines did not persecute
science, and they report nothing of the Inquisition. “Closing” the famous
Academy in Athens occurred quietly, as Gregorovius stated perplexedly. The
term “Hellenes” itself appeared very late in reliable history. According to
Gregorovius, it is only in the 15th century that Laonicus Chalcocondylas,
hailing from Athens, again christened his compatriots the “Hellenes” after
many hundreds of years of nonexistence [77]. Were the Hellenes, as stated by
traditional history, slavonized in Greece originally populated only by them,
or, on the contrary, were the Avars and Slavs living there in the late Middle
Ages hellenized? Theories of slavonizing the ancient Greeks are only based on
guesswork. On the other hand, the 10th-century Byzantine historian P. Safank
directly declares that almost all of Epirus and Hellas, the Peloponnese, and
Macedon were populated by the Scythian Slavs. Gregorovius continues that,
due to similar Byzantine evidence, the slavonization of the ancient Greek
lands should be accepted as a historical fact. The Slavonic names of towns,
rivers, mountains, and so on, densely covered medieval Greece (e.g., Goritza,
Krivitza, etc.), and it is only since the 13— 15th century that Greek Hellenic
names appeared and were later declared “ancient” .
Greece first (!) appears on the real political arena as a country of mutiny
and with a mixed, more than a half Slavonic population, only in the 8th
century. Nevertheless, again after empress Theophano has fallen, Athens, as
well as the rest of Hellas, disappears from the historical arena, so that it is even
hard to come upon a mention of the city. And only the Peloponnese, where
the Slavs settled most firmly, was a pretext for the Byzantines to interfere
with Greek affairs. There is still very little information about Greece in the
8-10th century. Gregorovius declares that neither history nor legend violates
the silence enshrouding the fate of the glorious city. This absence of facts
is so complete that those who investigate the signs of life (! — A.F.) in the
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famous city during the centuries described will be glad to encounter, as if a
discovery, at least some most insignificant detail similar to that given in St.
Luke’s biography about a thaumaturgist’s visit to Athens [77].
Greece and Athens emerge from darkness only in the 15th century. Greece
becomes especially important during the Crusades in the 12— 13th century.
Possessing a good port, and being a Venetian ally (there are many reasons to
believe that Venice should be identified with Phoenicia), Athens advances to
one of the top positions. It is important that chronological dates were indi-
cated in Greece with reference to the Christian era and in Arabic numerals
starting only in a.d. 1600. Gregorovius adds that the influence of time and
climate made decoding of these few inscriptions very complicated. They did
not even cast any light on Athenian history in the Christian era. The inves-
tigator of the medieval Roman past finds himself in an uncomparably more
advantageous position (we have already spoken of Rome). The annals of the
dead, hewn in stone, are totally absent in Athens. Few tombstones, one or
two sarcophagi without any statues, plus several inscriptions are all that is
left from the past (not counting the so-called ancient ruins). There are several
contradictory versions in traditional history regarding Athens in the 12-14th
century. According to one, the city and Greece were still in the dark. According
to another, Athens gradually started acquiring importance as a big cultural
centre [77]. For example, British scientists studied there. The Crusades were
not so much great religious and military ventures as important secular events.
The expeditions were headed by high European nobility (see the lists in [77]).
Greek territory was converted into the mosaics of feudal states whose role
is estimated today essentially from the negative point of view. It is assumed
that cruel and ignorant conquerors buried the great heredity of Greece. On
the other hand, Gregorovius (just having accused the Crusaders of vandal-
ism) unexpectedly informs us that its new history was just discovered by the
Latins, and turned out to be as multifarious as ancient. The Venetian nobiles
lusting for adventures set for the Greek seas, turning themselves into the 13th-
century Argonauts (later described by “ancient” poets — A.F.) [77]. Although
the history of the Frankish Crusaders’ 13- 14th-century states in Greece is
known fragmentarily, there was a time when fairy tales and legends turned
into life. The House of Villehardouin Gottfried II was famed even in the West
as a school of most refined customs. The Genoese merchants stayed in Thebes
and Athens and successfully competed with the Venetians. This was the era of
a remarkable golden age for literature and the arts, of which, though, almost
nothing remains. In my opinion, the 13— 14th century was the only epoch of
“ancient” Greece, which ended in 1453 with the conquering of the Byzantine
Empire by the Arabs. As Gregorovius puts it, the situation of the Frankish
states in Greece could not be called favourable at the beginning of the 14th
century. The Latins led the brilliant life of knights, which can be proved by
the convening of the parliament in May 1305 in Corinth. The isthmus, where
in the times of old the Poseidon games were staged, has become the arena of
knights’ tournaments in honour of beautiful women [77].
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191
It is important that the Frankish barons decorated their palaces with Greek
inscriptions (!). Adherents to historical tradition themselves stress a great
deal of “parallels” between the medieval and ancient Greek states. We will
not be able to give their list here, for this will necessitate a chain of extensive
tables made while investigating the GCD. We shall illustrate this only by one
representative example. Dante’s contemporary, Ramon Muntaner, recounted
the following event, absolutely neglecting the fact that this event and its
dating strongly contradicted traditional chronology and history (though this
was established only after his death). According to Muntaner, one Trojan
outpost was situated on Atracia, a promontory in Asia Minor, not far from
the island of Tenedos, where noble men and women of Romagna used to revere
a divinity. When Helen, the consort of the duke of Athens, went to that place
accompanied by hundreds of knights, she encountered the Trojan king’s son
Paris, who killed every one of her suite of knights and abducted the beauty
[77]. I advise the reader to refer to the GCD to discover that the “original”
of the famous Trojan war occurred, in fact, in the middle of the 13th century
in Italy or at the beginning of the 13th century in Constantinople.
It is important that the history of the Greek Frankish states was first studied
only in the 19th century. According to W. Muller, the archives give us only
the plot of this romantic drama whose theatre had been Greece for 250 years
from the 13th to the 15th century. In the 13th century, the Parthenon served
as a “Latin” temple of the Virgin Mary of Athens, “as if it had just been built”
(W. Muller) [1]. The famous statue of the Catholic Virgin Mary is placed in
the Parthenon as a copy (!) of the world-known statue of the pagan Athene by
Phidias (whose loss has been deplored by traditional history). The statue was
made in the 13th century. Another “ancient” temple devoted to the virgin,
and now called the Erechteon, was also built in the 13th century and is still
acting as if it has just been built, and so forth ([1], Vol. 4).
Gregorovius tells us about the famous Byzantine George Gemistus (Pletho),
“a resurrected ancient Hellene” living at the court of Theodore II. George was
a great worshipper of ancient gods. It was just at that time that the “Hel-
lenistic idea” of calling for the medieval Greeks to unite against conquerors
gained ground [77].
Archaeology in Athens began in a.d. 1447, i.e., when Cyriacus of Ancona
appeared in the city. He was the first to introduce the world of ruins of Athens
to the field of Western science. Cyriacus compiled the first catalogue of in-
scriptions and local names of monuments. These documents were lost, but
the contemporary historians are familiar with his data from a rendering by
15- 16th- century authors [77].
According to Gregorovius, the original names of most ancient Athenian
monuments now lying in ruins, were forgotten in the course of time. The fan-
tasy of antiquity lovers was anxious to relate them with the names of outstand-
ing men of the past. The remains of the Olympieion were called a “basilica”
in those centuries because, according to Gregorovius, nobody (!) knew that
these were the ruins of the formerly world-famous temple of Zeus at Olympia.
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Cyriacus called these vast ruins the palace of Adrian as did the citizens of
Athens themselves (who, therefore, erred, and only the later historians found
the truth and “corrected” the mistake). As early as 1672, Babin did not know
where in Athens the temple of Zeus was located. Several years later, J. Spon
was just as bewildered. The ruins of the Stoa were attributed to belong to the
palaces of Themistocles or Pericles; within the walls of the Odeum, the palace
of Herodes Atticus was thought to belong to Miltiades, and other ruins of
unknown buildings were assumed to be the houses of Solon, Thycidides, and
Aicmaeon. In 1647, Poentelle was shown the ancient ruins of Pericles’ palace,
and the Tower of the Winds was called “Socrates’ tomb”. The remembrance
of Demosthenes was related to the Monument of Lysicrates. This monument
was called “Demosthenes’ lamp” . The Academia, the Lyceum, the Stoa, and
Epicurus’ garden vanished without leaving a trace. In the times of Cyriacus,
one group of basilicas, or vast ruins whose foundations are now impossible to
find, were called the “Academia”. The didascalia of Plato “in the garden” was
also shown; it seems to be a tower in the Gardens of Ampelocypi. Rumours
about some of Caisarini’s schools on this mount circulated. The lyceum or
didascalia of Aristotle was placed in the ruins of Dionysius* theatre. The Stoa
and Epicurus’ school were even transferred to the Acropolis, into those big
buildings which were, probably, part of the Propylaea, and the temple of Nike
seems to have been the one taken over by the Pythagorean school.
We do not continue, because the state of archaeological chaos has now be-
come clear, and the list occupies several pages. To think that all this happened
in the 16th— 17th century!
The Byzantine Empire fell in 1453. The last Franks defended the Acropolis
for some time; however, infuriated by the stubborn resistance of this strong
fortress, Omar ordered artillery to shell (!) the Acropolis and its surroundings,
due to which its temples, and the Acropolis itself, were turned into ruins [77].
The powerful destruction of many a wonderful monument of the crusaders’
epoch led to the Athenian ruins then declared to be “ancient” .
Gregorovius writes that after the 15th-century Turkish invasion, Athens
was again (and how many times it was!) immersed into darkness. During the
Turkish yoke, the historian of Athens and Greece faces a problem as difficult as
ungratifying. He sees a desert before him [77]. The West has accepted the fall
of Greece, and almost forgot it. A German humanist confined himself to a note
in 1493 that the city of Athens was most glorious in Attica, from which only
few traces remained. In the 16th century, it became necessary for science to
possess exact information regarding the fate of the glorious city, which found
its expression in the problem whether Athens existed at all. The question
was raised by one German philhellene, Martinus Crusius. By rediscovering
Athens he made himself immortal. He sent a letter to Zugomalas Theodosios,
chancellor of the patriarch of Constantinople in 1573, asking to inform him
whether the mother of all knowledge, as stated by the German historians,
does exist, that the city of Athens was effaced, with just a few sailors’ huts in
its place. The reply of the enlightened Byzantine together with the later letter
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The “Dark Ages ” in Medieval History
193
of an Acharnian were the first exact data which calmed the German scientist
as regards the existence of the city. For the first time, they cast a dim light
on the state of its monuments and the flora of its people (in which, according
to traditional history, the legend is enrooted that the Parthenon was erected
by the architects Ictinus and Callicrates under the statesman and strategos
Pericles, the popular leader of the democratic party created in Athens as far
back as the 5th century B.c. and who died of the plague in 429 B.C. together
with its leader, although it is unknown in which month — A.F.) [77].
The scientific archaeology of Athens started only in the 17th century when
Scaliger’s chronology had already been created with the works of Jean le Maire
from the Netherlands [77]. Nevertheless — let me refer to Gregorovius again —
even in 1835, one German scientist expressed the opinion that an uninhabited
desert had remained in place of Athens for four hundred years. Compared
with the study of the city of Rome, the archaeology of Athens was about two
centuries late.
The prejudice firmly enrooted in Europe that Athens did not exist, as it
were, could be eliminated only with one’s own eyes: This was to the credit of
the French Jesuits and Capuchins, who first appeared in Athens in 1645 [77].
In the second half of the 17th century, the French monks made the first (!)
maps of the city. The continuous and more or less scientific study of Athens
began only at this moment when traditional chronology had been almost
created, and the Greek monuments had already been dated on the basis of
the distorted chronology of Rome, which also led to lengthening Greek history
artificially.
14.3. The history of religions
In conclusion, we shall briefly discuss the situation of the history of religions.
It is assumed traditionally that each chronological epoch has had its own
religious cults separated by centuries and millennia. At the same time, the
19th-century historians and ethnographers did the enormous job of compar-
atively studying the world religions and cults. They found that the religions
traditionally believed to be separated by hundreds and thousands of years
admit an extraordinary large number of “parallels” (sometimes even identical
coincidences). This fact engendered numerous theories of influence, borrow-
ing, infiltration, and so forth. However, they all rest on traditional chronology
and are generated only by it. A change of chronology will make the scientists
revise the prior point of view. Since we do not have the space here, we only
indicate some typical examples:
A so-called Celtic monument found in 1771 is regarded as the traditional
representation of the pagan pre-Christian forest-god of the Gauls [37]. How-
ever, the inscription ESUS carved out over the divinity’s head could be seen
explicitly. Nevertheless, traditional chronology makes the historians believe
that this is the pre-Christian “God Jesus”. The well-known historian and spe-
cialist in comparative religion A. Drews wrote that he considered mythological
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New Experimental and Statistical Methods
Chapter 3
parallels between Christianity and paganism very important. Those who do
not see the generally known relation of the Gospel's paschal history to the
myths and cults of the religion venerating Attis-Adonis-Osiris, and others,
those who affirm that the myths of Attis and Adonis say nothing of Sunday
burials, who hope to prove that Jesus’ death was different from the deaths of
his relatives in Asia Minor, who cannot recognize the Virgin Mary in the nu-
merous Indian, Anatolian, and Egyptian mother-goddesses Maia, Mariamma,
the mother of the “Messiah” Cyrus, weeping Semiramis, Myrrha, and Mera,
should be left out of the discussion of questions of religion and history. Drews
lists a great many parallels identifying the Christian “holy family” with other
“holy families” of the Anatolian and Egyptian gods allegedly separated from
the turn of the millennium by many centuries. Rejecting the traditional ap-
proach, we see that all the parallels simply point to the simultaneity of these
cults, which are different only in national character because of their origin.
For example, the principal holy shrine of Mithras had been in the Vatican
in the place of today’s San Pietro. Mithras-Attis was then called “father’s
father”. The high priest serving this god was also called “father” (father’s
father) just as the Roman pope is still called the “Holy Father” [82]. Mithraism
as well as Christianity accepts the tenet of purgatory, use of a holy-water
basin, and the ritual of crossing oneself. The rituals of public service, Mass,
host, wafer of consecration, communion bread, and so forth, are perfectly the
same [82]. Mithras’ and Christian cults are practically indistinguishable, and
the difference between them in hundreds of years is noted only by traditional
chronology. It turns out that the mixed cult of the Egyptian goddess Isis,
whose worshippers had matins, liturgies, and vespers surprisingly similar to
the Catholic and even partly Orthodox religious services, is nearly coincident
with the medieval Christian cult [82] .
Without putting traditional chronology to doubt, which has moved the
Isis-Osiris-Serapis cult into hoary antiquity, the historian of religions N. V.
Rumyantsev had to declare that
44 . . . this coincidence of prayers of the Egyptian religious service with Christian
liturgical prayers is too complete and striking to be accidental’* ([83], p. 72).
The “land of crosses” is traditionally believed to be ancient Egypt. The same
Christian crosses were spread in ancient India, Mesopotamia, and Persia. A
great deal of the Egyptian images of gods contained the anagram of Christ
[83. Summarizing his study, Rumyantsev wrote:
44 A series of suffering, dying and resurrecting gods of the ancient world have passed
before us; we saw their myths, grew familiar with the holidays, rituals, and so forth,
devoted to them. In spite of the difference in their names, myths, birthplaces or histor-
ical arena, it can still be felt, even against one’s own will, that they all have something
in common. Moreover, the people of antiquity noticed this fact, too. ... In fact, if we
look at the recent centuries before and after the so-called Nativity, then we shall notice
a curious thing. All the divinities listed above have been closely linked with everything
related to them, even sometimes indistinguishably. Osiris, Tammuz, Attis, Dionysius,
and others, formed some unique, conunon, and conjoint image, thus creating a certain
syncretic (mixed) divinity almost undividedly reigning over the whole vast territory of
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The “ Dark Ages” in Medieval History
195
the Roman state The divinities, in fact, turned into one Savior with many names.
This tight conjunction occurred especially in the epoch of the Roman Empire and, in
particular, in Rome itself” ([S3], pp. 44-45).
Egyptian chronology yawns with enormous gaps and is a set of separate
fragments quite unrelated to each other or even completely independent. As
in the history of Europe, the “restoration” effect can be made manifest in
Egyptian history during the Saite restoration when the cults, customs, written
language, and so on, forgotten long ago were revived again. Being intimately
related with Biblical and Roman chronology, the whole of Egyptian history
also undergoes “glueing together” (of “different” Egyptian epochs) and “com-
pression”. As a result, the ancient and medieval Egyptian history became
shorter, similarly to the history of Europe (see the GCD). The forward shift
is consistent with the results obtained independently even by Isaac Newton
while investigating the chronology of Egypt [67].
14.4. Indian history and chronology
Oriental history is also linked with that of Europe and Egypt. For example,
let us give a short quotation by N. Guseva pertaining to the chronology of
India:
“The science of history faces in India such difficulties that cannot even be imagined
by the investigators studying the ancient history of other countries and peoples (this
was written in 1968 — A.F.). The most difficult one of them is the complete absence
of dated sources” (see the Russian translation of D. Kosambi’s book The Culture and
Civilization oj Ancient India in Historical Outline [84]).
All the basic chronological milestones in Indian history were established by
the comparison with Roman, Greek, and Egyptian chronology. D. Kosambi
writes:
“India has virtually no historical records worth the name. ... In India there is only
vague popular tradition, with very little documentation above the level of myth and
legend. We cannot reconstruct a complete list of kings. . . . What little is left is so
nebulous that virtually no dates can be determined for any Indian personality till the
Muslim period This has led otherwise intelligent scholars to state that India has
no history” ([84], pp. 9-10).
The medieval authors sometimes placed India in Africa and even in Italy
(!). Similarly to Europe, at the beginning of the millennium, India “suddenly”
happens to be at a “barbaric stage of its development” , again starting its way
to the top level of civilization [84]. The golden age of the Sanskrit Indian
literature is dated only to the 11th century! Indian medieval history is also
rich in chronological gaps as long as a century, being intricate and chaotic.
“Thus, brahmin indifference to past and present reality . . . only erased Indian
history. . . . For historical descriptions of ancient Indian scenes and people ... we have
to rely upon Greek geographers, Arab merchant travellers. . . . Not one Indian source
exists of comparable value” ([84], pp. 174-175).
Thus, Indian chronology and history are completely dependent upon those
of Rome and Greece and are restructured following the former.
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Chapter 3
Remark. I cannot at all agree with the hypothesis of Morozov, according
to which most literary works of antiquity are fabrications of the Apocrypha
of the Renaissance, which would mean that what we know today as ancient
history is actually the result of premeditated falsification. This thesis for-
mulated in [1] caused justified criticism. My standpoint is different, namely
that, due to the results of the application of the new dating methods (see
above), almost all surviving ancient documents (of antiquity or the Middle
Ages) are authentic and were written for the purpose of perpetuating real
events rather than leading future historians astray. More than that, certain
of numerous examples of the GCD and its decomposition (and also the new
version of chronology I suggested) justifying the authenticity of many a doc-
ument (e.g., of the Donation of Constantine , the Almagest of Ptolemy, etc.)
were given above, i.e., many of the documents regarded today as adulterated
turn out to be originals, which are extremely consistent with the new version
of chronology, following from the GCD and its decomposition into the sum of
three shifts. For example, this refers to the “privileges” given by Caesar and
Nero to the Austrian duchy (see above). In my opinion, practically everything
described in the old documents did, in fact, occur. The problem is when and
where? The confusion discovered earlier, which led to the lengthening of au-
thentic history due to the natural chronological error (e.g., because of mixing
up the dates of the foundation of the “two Romes”), only took place in solv-
ing it. The new version of chronology which I have suggested (and which is
essentially different not only from the traditional, but also from Morozov’s
version) makes us redate the old documents and does not at all deny their
validity as true witnesses of the past events.
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Index
A
Aconiates, Nicetas, 148
Acropolita, Georgius, 148
Africanus, 24, 139
Alberti, L., 98
Almagest , 114
Ancient eclipses, 15
Ancient maps, 160
de Arcilla, 4, 96
Athribis horoscopes, 34, 35
Avignon exile, 172
B
Babylonian captivity, 172
Baldauf, R., 96
Barzizza, 97
Bethlehem, star of, 32
Bible, 152, 155
Biblical geography, 104, 107
Biblical history, 168
Bickerman, E., 4
Blair, J., 55
Bracciolioni, P., 97
Brahe, Tycho, 130
Brugsch, H., 3, 95, 138
C
Cicero, 97
Cinnamus, Johannes, 148
Christian chronographers, 90
Chronological shifts, 166
Codex Alexandrinus, 102
Codex Sinaiticus, 102
Codex Vaticanus, 102
Comnena, Anna, 148
Correlation of local maxima,
Council of Trent, 178
D
Dendrochronology, 133
Dependent dynasties, 68, 73, 163
Dependent texts, 7, 47, 141
Duplicates, 8
Duplicates in the Bible, 157
Duplicate recognition, 8, 67, 79
Duplication effect, 7
Durer, Albrecht, 114, 116
Dynastic stream, 55
Dynasty of rulers, 68
E
Enquete-code, 57, 159
Epoch-duplicates, 83
Euclid, 99
Eusebius Pamphili, 90, 139
F
Flamsteed, J., 130
Flinders Petrie, 34
Frequency- damping principle, 76, 145
Frequency-duplicating principle, 79, 151
Frequency histogram, 70
G
Gestorum Pontificum Romanorum, 148
Ginzel, F., 15, 17, 18
Global chronology diagram (GCD), 8,
30, 63-65, 73, 162
Graph of D", 2, 26, 27
Gregoras, Nichephorus, 148
Gregorovius, F., 53, 148
H
Halley, E., 128
Hardouin, J., 4, 96
Herodotus, 107
43 Hipparchus, 114
Hochart, P., 97
203
204
Index
Horoscopes, 34, 35, 136
Hypercriticism, 93
K
Kepler, J., 33
I
Ideler, L., 33
Independent dynasties, 68, 141
Independent texts, 7
Information-loss model, 6
Informative function, 41
L
Libby, W., 134
Liber Pontificalis, 148
Livy, T., 22, 23, 32, 49, 148
Lubieniecki, Stanislaw, 115
M
Machiavelli, 152
Martynov, G., 94
Maxima correlation principle, 7, 47, 49
Medieval anachronisms, 99
Medieval duplicates, 110
Method of ordering texts, 145
Milojcic, Vladimir, 135
Mommsen, T., 3, 93, 148
Morozov, N.A., 4, 17, 34, 88, 89, 113,
136, 174
Moon's elongation, 1, 12, 15
N
Nash papyrus, 103
Neugebauer, 0 M 96
P
Pachymeres, Georgius, 148
Petavius, D., 4, 90, 174, 178
Phlegon, 24, 139
Pletho, G., 110
Plotinus, 110
Plutarch, 150
Pompeii, 131
Primary stock, 5
Psellus, Michael, 148
Ptolemy, 114
Q
Qumran mauscripts, 103
R
Radiocarbon method, 133
Renan, E., 139
Revelation, Book of, 35, 81
Roman chronology, 89
Ross, J., 97
S
Second derivative of the moon's
elongation, 15, 16
Sergeev, V,S., 54, 142
Scaliger, J., 4, 90, 92, 178
SmalLdistortion principle, 68, 72, 143
Star catalogue, 115
Statistical dating methods, 5, 39, 140
Sunderland, J., 139
Synkellos, 24, 139
T
Tacitus, Cornelius, 97
Thucydides, 20, 21, 31
Tischendorf, L., 102
Turaev, B.A., 3
Newton, I., 4, 96
Newton, R., 1, 15, 26, 115
Newton's (R.) conjecture, 1
Numerical dynasties, 68, 144
Numismatics, 132
O
Old Testament, 168
Oppolzer, T., 17, 18
Vilyev, M.A., 34
Virtual dynasties, 70
Vitruvius, 98
Volume graph, 5, 140
Volume function, 40
W
Wooley, L., 96
CONTENTS OF VOLUME II
Introduction xiii
Chapter 1. Methods for the Statistical Analysis of Narrative Texts 1
1. The Maximum Correlation Principle for Historical Chronicles and Its Ver-
ification by Distribution Functions. Analysis of Russian Chronicles 1
2. The Maximum Correlation Principle and Its Verification by Frequency His-
tograms. Method for the Discovery of Dependent Historical Texts. The
Period of “Confusion” in the History of Russia (1584-1600 A. D.) 3
3. A Method for Dating Historical Events Described in Chronographic Texts,
and Its Verification Against Reliable Historical Data 9
4. Methods for Ordering and Dating Old Geographic Maps and Descriptions 11
4.1. The map-code and the map-improvement principle 11
4.2. Confirmation of the map-improvement principle 15
4.3. Herodotus' map 16
4.4. Medieval geography 17
5. Frequency Distributions in Rulers' Numerical Dynasties 19
5.1. Parallel rulers' dynasties 19
5.2. Statistical parallel between the Carolingians and the Third Roman
Empire 23
5.3. Statistical parallel between the Holy Roman Empire and the Third
Roman Empire 23
5.4. Statistical parallel between the Holy Roman Empire and the Empire
of the House of Hapsburg 26
5.5. Statistical parallel between the Holy Roman Empire and the Second
Roman Empire 29
5.6. Statistical parallel between the Holy Roman Empire and the kingdom
of Judah 32
5.7. Statistical parallel between Roman coronations of the Holy Roman
emperors and the kingdom of Israel 34
5.8. Statistical parallel between the First Roman pontificate and the Sec-
ond Roman pontificate 39
Contents
5.9. Statistical parallel between the First Roman Empire (regal Rome)
and the Third Roman Empire 41
5.10. Statistical parallel between the Second Roman Empire and the Third
Roman Empire 47
5.11. Statistical parallel between the kingdom of Judah and the Eastern
Roman Empire 48
5.12. Statistical parallel between the kingdom of Israel and the Third Ro-
man Empire 49
5.13. Statistical parallel between the First Byzantine Empire and the Sec-
ond Byzantine Empire 52
5.14. Statistical parallel between the Second Byzantine Empire and the
Third Byzantine Empire 53
5.15. Statistical parallel between medieval Greece and ancient Greece 54
5.16. Statistical duplicates of the Trojan war 57
5.17. “Modern textbook of European history” and its decomposition into
the sum of four short isomorphic chronicles 58
5.18. Possible explanation of the three chronological shifts discovered in the
Global Chronological Diagram 71
1. The general idea and the 1,000-year shift 71
2. The 333-year shift 73
3. The 1,800-year shift 74
5.19. Dionysius the Little 77
5. Some Other Independent Proofs of the Existence of Three Basic GCD
Chronological Shifts 78
6.1. The list of Roman popes as the spinal column of medieval Roman
history 78
6.2. The mean age of all old historical names and the frequency-damping
principle for the matrix columns 82
6.3. Square matrix of biblical names and statistical duplicates in the Old
and New Testament 85
6.4. Matrix of parallel passages in the Old and New Testament 87
6.5. Scatterings of related names in chronological lists. The relation ma-
trix 90
1. Introduction 90
2. Name list of secular or church rulers 91
3. Correct and incorrect chronology in the name list. Frequency
histograms 92
4. Computation of histograms for real historical texts 93
5. Histograms related to the name and nationality lists of Roman
popes 94
6. Damping succession in a historical chronicle 95
7. Results related to the lists of biblical names and parallel passages 97
Contents
8. Chronological shifts between the duplicates in chronologically in-
correct chronicles 98
9. The card-deck problem and chronology 100
10. Relation matrix: preliminaries 101
11. Principal definitions. Assumptions about the structure of a cor-
rect chronological text 101
12. Relation measure. The problem of separation of strong and weak
relations in a chronicle 103
13. Frequency histograms for the appearance of relations. The choice of
thresholds 105
14. Results related to the name list of Roman popes. Chronological
shifts 106
15. The list of names of Roman emperors and the related chronolog-
ical shifts 110
16. The comparison of the results obtained with the decomposition
in the Global Chronological Diagram 111
Chapter 2. Enquete-Codes of Chronological Duplicates and Biographical Par-
allels. Three Chronological Shifts: The Byzantine-Roman 333-year shift, the
Roman 1,053-year shift and the Greco-biblical 1,800-year shift 112
1 . Frequency Characteristics and Enquete-Codes of the Historical Periods
from 82 B.C. to 217 A.D. (Second Roman Empire) and from 300 to 550
A.D. (Third Roman Empire). The 330-year First Basic Rigid Shift in
Roman History 112
1.1. Ancient sources and their origin. Tacitus and Bracciolini 112
1 .2. The complete list of Roman emperors of the Second and Third Roman
Empires 118
1.3. The 330-year rigid shift in Roman history. The parallel between the
Second and the Third Roman Empires. Remarkable Biographical
Parallels 124
2. Charlemagne’s Empire and the Byzantine Empire. The 330-year Rigid
Shift. Comparison of the 4-6th cc. A.D. and the 7-9th cc. A.D. 144
3. Chronological “Cut” in the Traditional Version of Ancient History 149
4. The 1,053-year Second Basic Chronological Shift in European History 152
4.1. The general structure of the 1,053-year second chronological shift and
the 1,800-year third chronological shift 152
4.2. The formula of the shift X + 300. Parallels between the First Roman
Empire (Regal Rome), the Third Roman Empire and the Bible. The
first 250 years of Roman history 154
4.3. War against the Tarquins and the Gothic war. The 1,053-year chrono-
logical shift and the formula X + 300. Comparison of the historical
events of the 6th c. B.C. and the 6th c. A.D. 163
1. War prehistory 164
2. Start of the GTR-war 166
Contents
3. War with Rome 168
4. Stream of parallel events 171
5. End of the GTR-war 176
4.4. The Second Roman Empire and the Holy Roman Empire in the 10-
13th cc. A.D. The 1,053-year chronological shift and the formula
X + 300 177
1. Ancient Rome and medieval Rome in 555-850 A.D. 177
2. John the Baptist and John Crescentius (10th c. A.D.) 178
3. Jesus Christ and Gregory VII Hildebrand (11th c. A.D.) 183
4. Star flares in the Second Roman Empire and the Holy Roman
Empire. The “evangelical star” in 1 A.D. and star flare in 1054
A.D. 189
5. Eclipse that occured during the Crucifixion 190
4.5. The Third Roman Empire and the Holy Roman Empire. The 720-
year chronological shift as the difference between the first and second
basic chronological shifts. The Trojan war, Gothic war and Italian
war in the 13th c.A.D. 194
5. The Parallel between the Western Third Roman Empire and the Biblical
Kings of Israel. Enquete-Codes of the Historical Periods of the 9-5th cc.
B.C. and the 3rd-6th cc. A.D. 197
5.1. The complete table of both streams 197
5.2. The remarkable biographical parallel 200
6. The Parallel between the Eastern Third Roman Empire and the Biblical
Kingdom of Judah 214
6.1. The complete table of both streams 214
6.2. A remarkable biographical parallel 215
7. The Medieval Song of Roland and the Biblical Book of Joshua 226
7.1. History of the poem “Song of Roland” 226
7.2. The parallel between the medieval poem and the ancient chronicle.
Table of the isomorphisms 228
8. The 1,800-year Third Basic Rigid Shift in Ancient Chronology. The Gothic
= Trojan = Tarquins’ War (= GTR war) and Its Chronological Duplicates
in the Different Epochs of Traditional History 233
8.1. The Trojan war and the Gothic and Tarquinian wars 233
1. The medieval Trojan cycle. Homer, Dares and Dictys 233
2. A rough comparison 235
3. The “legend of a woman” and the start of war 243
4. The fall of Naples and Troy 246
5. The Greeks’ Trojan horse and the Latins’ aqueduct of Naples 247
6. Achilles and Patroclus = Valerius and Brutus 250
Contents
7. Achilles and Hector = Belisarius and Vitiges 252
8. Achilles’ “betrayal” and Belisarius’ “betrayal” 254
9. Troilus = Totila; Paris = Porsena 256
10. The other Trojan legends 259
11. Medieval anachronism in the ancient Ttojan cycle 262
12. The Christian dating of the Trojan war 264
8.2. The Reflection of the Trojan war and the GTR-war in the 1st c. B.C.
(Sulla, Pompey and Julius Caesar) 266
1. New parallels in Roman history (the “great Triumvirate” : Sulla,
Pompey, Julius Caesar and the GTR-war in the 6th c. A.D.) 266
2. Four statistical duplicates: the Gothic war in 6th c. A.D. = the
Roman war (Julius Caesar) in 1st c. B.C. = the Trojan war in
the 13th c. B.C. and = the Tarquinan war in the 6th c. B.C. 267
3. The “principal king” : Justinian = Pompey = Agamemnon = Tar-
quinius the Proud 269
4. The “legend of a woman” 270
5. Marcius Junius Brutus and Patroclus 274
6. Vercingetorix and Hector 277
7. Julius Caesar and Achilles 281
8. Anthony and Antonina 285
8.3. The GTR-war of the 6th c. A.D. and the Nika riot of the 6th c. A.D. 288
9. Egyptian Chronology 291
9.1. Difficulties in creating Egyptian chronology 291
9.2. Astronomical dating of the zodiacs in the temple in Dandarach 293
1. The “round zodiac” and its horoscope. History of the problem 293
2. Decoding of the “round” zodiac by comparing it with medieval
astrographic charts 296
10. Some Strange Features of Ptolemy’s Almagest. Preliminary Remarks 303
10.1. Latin and Greek editions 303
10.2. Diirer’s astrographic charts in the first editions of the Almagest 306
11. Duplicates in Greek Chronology. The 1,800-year Chronological Shift 308
11.1. The Epoch of the Crusades in 1099-1230 A.D. and the Epoch of the
Great Greek Colonization in the 8-6th cc. B.C. 308
11.2. Charles of Anjou and Cyrus 311
11.3. Matilda and Miltiades 313
11.4. The Greco-Persian war and the battle of 300 Spartans with Xerxes’
armies at Thermopylae 315
11.5. The war in medieval Greece and the Peloponnesian war in ancient
Greece 316
Contents
11.6. The medieval Mahometans and the ancient Macedonians.
Mahomet II and Philip II 319
Appendix 1. Volume Graphs for the “Biographies” of the Holy Roman Em-
perors of the 10-13th cc. A.D. Additional Chronological and Statistical Data
of Ancient History 323
Appendix 2. When Was Ptolemy’s Star Catalogue Really Compiled? Vari-
able Configurations of the Stars and the Astronomical Dating of the Almagest
Star Catalogue 346
1. History of the Problem and Subject of the Work 346
2. Some Notions from Astronomy 347
3. Some Characteristics of the Ancient Star Catalogues 349
4. Errors in the Coordinates in Ancient Catalogues 350
5. Preliminary Analysis of the Almagest 351
6. General Description of the Method of Dating 354
6.1. Types of errors occurring in the catalogues 354
6.2. Systematic errors 354
6.3. Random errors and spikes 356
7. Statistical Analysis of the Almagest Star Catalogue 357
7.1. Preliminary remarks 357
7.2. Classification of latitude errors 357
7.3. Analysis of errors. Seven homogeneous regions in the Almagest star
atlas 359
7.4. Error values in the Almagest star catalogue 363
8. The Dating of the Almagest Star Catalogue 366
8.1. Statistical dating procedure 366
8.2. Geometrical dating procedure 366
9.. Stability of the Method 368
10. Dating of Other Catalogues 369
10.1. Tycho Brahe’s catalogue 369
10.2. Hevelius’ catalogue 373
10.3. Ulugbeck’s catalogue 374
10.4. Al-sufi’s catalogue 375
Appendix 3. Dating of the Almagest Based on the Occultation of the Stars
by Planets and Lunar Eclipses 376
1. Introduction 376
Contents
2. Dating of the Occultation of the Stars by Planets 377
3. Dating of the Lunar Eclipses 382
4. The Chronology of the Almagest 387
Appendix 4. The Dating of the First Oecumenical Council of Nicaea and the
Beginning of the Christian Era 390
1. A date for the Council of Nicaea from the Easter Book 390
1.1. The accepted point of view 390
1.2. A date from the Easter determination rule. A computer experiment 393
1.3. A date from Easter full moons 394
1.4. A date from the “Damaskine palm” 395
1.5. An explicit date of Matthew Vlastar 396
1.6. Comparison of the dates 396
1.7. The “first and second” Oecumenical Council. Canonization of the
Easter Book 397
1.8. The Gregorian calendar reform 398
1.9. Where the date for the Council of Nicaea came from 400
1.10. The main conclusions 401
2. The Birth of Christ and the 1 A.D. 401
2.1. History of the problem 401
2.2. The “First Easter conditions” 402
2.3. A date for the first Easter from the complete set of the first Easter
conditions 403
2.4. Dates for the First Easter from the reduced set of the First Easter
conditions 404
2.5. On the lifetime of Dionysius Exiguus 404
3. On modern tradition 406
3.1. The extremity of modern dating (“the more ancient the better”) 406
3.2. Matthew Vlastar’s equinoxes and modern chronological tradition 408
Appendix 5. The Well-known Babylonian Captivity and the Well-known Avi-
gnon Exile of Papacy 412
Bibliography 421
Subject Index 437
Index of Names 441
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